WEBVTT 00:00:00.816 --> 00:00:04.206 A:middle >> Thank you for joining us for the Cypress College Math Review 00:00:04.496 --> 00:00:07.986 A:middle on arithmetic and geometric sequences. 00:00:08.906 --> 00:00:12.006 A:middle Objective one, sequences in general. 00:00:13.166 --> 00:00:16.316 A:middle A sequence is an ordered collection 00:00:16.416 --> 00:00:18.126 A:middle of numbers or objects. 00:00:18.906 --> 00:00:24.006 A:middle The notation that's commonly used is a sub 1 is the first 00:00:24.006 --> 00:00:29.036 A:middle term, a sub 2 is the second term, a sub 3 is the third term, 00:00:29.506 --> 00:00:31.016 A:middle and so on and so forth. 00:00:31.016 --> 00:00:34.836 A:middle [variable] a sub n is the nth term or the general term. 00:00:36.936 --> 00:00:41.516 A:middle Example, describe the sequence and determine the next object. 00:00:41.636 --> 00:00:46.916 A:middle We have a triangle and a square and a pentagon and a hexagon. 00:00:47.236 --> 00:00:49.486 A:middle Let's see. 00:00:49.486 --> 00:00:52.066 A:middle Well, this is a sequence of what are called polygons. 00:00:52.906 --> 00:00:54.976 A:middle The first object has three sides. 00:00:54.976 --> 00:00:56.706 A:middle It's a polygon with three sides. 00:00:57.306 --> 00:00:59.166 A:middle Then we have one with four sides 00:00:59.166 --> 00:01:01.476 A:middle and then five sides and then six sides. 00:01:02.066 --> 00:01:04.076 A:middle So, the next object 00:01:04.076 --> 00:01:06.706 A:middle in the sequence should be a polygon with seven sides. 00:01:07.346 --> 00:01:09.466 A:middle So, this is a sequence of polygons, 00:01:10.116 --> 00:01:11.976 A:middle and each polygon has one more side. 00:01:20.096 --> 00:01:22.516 A:middle In this example, we're asked to describe the sequence 00:01:22.796 --> 00:01:23.926 A:middle and determine the next term. 00:01:24.316 --> 00:01:28.456 A:middle We have 1, 4, 9, 16, 25. 00:01:28.596 --> 00:01:34.986 A:middle I find it's often easier to determine the pattern by looking 00:01:34.986 --> 00:01:37.036 A:middle at the later numbers 00:01:37.036 --> 00:01:38.696 A:middle in the sequence rather than the first few. 00:01:39.106 --> 00:01:41.976 A:middle So, we look at 9, 16, 25. 00:01:42.566 --> 00:01:49.416 A:middle Hmm, well, 16 is 4 squared, and 25 is 5 squared. 00:01:50.336 --> 00:01:51.706 A:middle Does that work for the other numbers? 00:01:52.856 --> 00:01:54.386 A:middle Well, 9 is 3 squared. 00:01:55.356 --> 00:02:00.186 A:middle And 4 is 2 squared and 1 is 1 squared. 00:02:01.096 --> 00:02:05.186 A:middle Ah, so these are all perfect squares. 00:02:05.736 --> 00:02:07.466 A:middle So, that's describing the sequence. 00:02:07.466 --> 00:02:08.676 A:middle They're all perfect squares. 00:02:09.036 --> 00:02:10.826 A:middle Now, what would the next term be? 00:02:11.046 --> 00:02:13.476 A:middle So, we have 1 squared, 2 squared, 00:02:13.536 --> 00:02:16.006 A:middle 3 squared, 4 squared, 5 squared. 00:02:16.196 --> 00:02:20.306 A:middle So, the next number would be 6 squared, which is 36. 00:02:20.676 --> 00:02:22.766 A:middle So, the next term is 36. 00:02:24.016 --> 00:02:26.596 A:middle There's another thing that we can do for this pattern. 00:02:27.376 --> 00:02:29.266 A:middle We can actually determine a formula 00:02:29.266 --> 00:02:30.646 A:middle for the terms in this sequence. 00:02:31.536 --> 00:02:34.166 A:middle Let's use the notation we described earlier. 00:02:34.546 --> 00:02:37.056 A:middle So, a sub 1 is 1 squared, 00:02:37.056 --> 00:02:40.996 A:middle a sub 2 with the subscript 2 is 2 squared. 00:02:41.306 --> 00:02:43.656 A:middle [variable] a sub 3 is 3 squared. 00:02:43.656 --> 00:02:45.806 A:middle [variable] a sub 4 is 4 squared. 00:02:45.806 --> 00:02:47.886 A:middle [variable] a sub 5 is 5 squared. 00:02:48.236 --> 00:02:51.376 A:middle So, this number, the subscript matches the base. 00:02:51.966 --> 00:02:57.716 A:middle So, if we have a sub n for the general term in the sequence, 00:02:58.456 --> 00:03:01.736 A:middle that would be the base of this exponential expression. 00:03:02.106 --> 00:03:04.986 A:middle So, a sub n would be n squared. 00:03:04.986 --> 00:03:06.196 A:middle This is the formula 00:03:06.566 --> 00:03:09.586 A:middle that represents the nth term in the sequence. 00:03:09.586 --> 00:03:12.016 A:middle So, a sub n is equal to n squared. 00:03:12.286 --> 00:03:15.086 A:middle That's a formula that represents the terms in the sequence. 00:03:15.946 --> 00:03:19.516 A:middle Let's say we want to figure out then the 10th term. 00:03:19.736 --> 00:03:22.086 A:middle So, we use this formula. 00:03:22.626 --> 00:03:26.496 A:middle And now we want a sub 10, the 10th term in the sequence. 00:03:26.856 --> 00:03:31.616 A:middle So, a sub 10 would be 10 squared, and 10 squared is 100. 00:03:31.856 --> 00:03:33.926 A:middle The 10th term of the sequence is 100. 00:03:39.276 --> 00:03:48.136 A:middle This sequence has the number 1, 1, 2, 3, 5, 8, 13, 21, 34. 00:03:48.576 --> 00:03:50.446 A:middle What's the pattern here? 00:03:52.286 --> 00:03:55.436 A:middle Hmm, well, we have 3 plus 5 is 8. 00:03:56.736 --> 00:03:58.606 A:middle Five plus 8 is 13. 00:04:00.036 --> 00:04:01.726 A:middle Eight plus 13 is 21. 00:04:01.726 --> 00:04:02.766 A:middle That works, doesn't it? 00:04:04.426 --> 00:04:05.736 A:middle Let's check it out from the beginning. 00:04:06.136 --> 00:04:09.766 A:middle If we take these two numbers here, the first two terms 00:04:09.766 --> 00:04:11.776 A:middle in the sequence and we add them up, 00:04:12.276 --> 00:04:13.856 A:middle we get the third number, don't we? 00:04:13.986 --> 00:04:15.746 A:middle 1 plus 1 is 2. 00:04:15.746 --> 00:04:20.566 A:middle Let's try and see if that pattern works for all of them. 00:04:20.566 --> 00:04:22.646 A:middle 1 plus 2 is 3. 00:04:24.106 --> 00:04:27.046 A:middle 2 plus 3 is five. 00:04:27.526 --> 00:04:29.636 A:middle 3 plus 5 is 8. 00:04:30.716 --> 00:04:32.906 A:middle 5 plus 8 is 13. 00:04:33.336 --> 00:04:34.646 A:middle Yes, that pattern works. 00:04:35.546 --> 00:04:37.816 A:middle This is called the Fibonacci sequence. 00:04:38.306 --> 00:04:40.256 A:middle It's named after the guy that came up with it. 00:04:40.496 --> 00:04:42.766 A:middle It's a very special sequence. 00:04:43.186 --> 00:04:46.026 A:middle It actually occurs a lot in the study of botany, 00:04:46.546 --> 00:04:50.276 A:middle computer science, genetics and many other fields. 00:04:51.206 --> 00:04:54.106 A:middle Let's find the 11th term of this sequence. 00:04:54.866 --> 00:04:56.816 A:middle Let's list out the terms. 00:04:57.336 --> 00:05:00.166 A:middle So, the first term, a sub 1 is 1. 00:05:00.546 --> 00:05:03.166 A:middle [variable] a sub 2 is also 1. 00:05:04.166 --> 00:05:09.246 A:middle [variable] a sub 3 is 2, a sub 4 is 3 00:05:09.246 --> 00:05:10.366 A:middle because it's the fourth term. 00:05:11.026 --> 00:05:12.416 A:middle [variable] a sub 5 is 5. 00:05:12.416 --> 00:05:15.026 A:middle [variable] a sub 6 is 8. 00:05:15.956 --> 00:05:18.146 A:middle [variable] a sub 7 is 13. 00:05:18.676 --> 00:05:20.996 A:middle [variable] a sub 8 is 21. 00:05:22.356 --> 00:05:24.506 A:middle [variable] a sub 9 is 34. 00:05:24.506 --> 00:05:28.976 A:middle Now, how do we get a sub 10? 00:05:29.226 --> 00:05:30.976 A:middle Well, to get a sub 10, I would have 00:05:30.976 --> 00:05:33.756 A:middle to add the previous two terms of the sequence. 00:05:34.176 --> 00:05:37.626 A:middle 21 plus 34 is 55. 00:05:38.306 --> 00:05:40.136 A:middle So, a sub 10 is 55. 00:05:41.116 --> 00:05:47.316 A:middle [variable] a sub 11, I need to add 34 and 55 and we get 89. 00:05:47.816 --> 00:05:52.706 A:middle The 11th term of the Fibonacci sequence is 89. 00:05:59.046 --> 00:06:02.656 A:middle In this problem, we're given the formula, a sub n equals n 00:06:02.656 --> 00:06:04.376 A:middle over the quantity n plus two. 00:06:05.076 --> 00:06:07.076 A:middle And we're supposed to find the first four terms. 00:06:07.256 --> 00:06:09.266 A:middle What does that mean, find the first four terms? 00:06:09.896 --> 00:06:14.746 A:middle It means to find a sub 1, which means plug in 1 for n, 00:06:14.746 --> 00:06:16.926 A:middle and then find a sub 2, and so on and so forth. 00:06:17.396 --> 00:06:19.016 A:middle So, here's our formula. 00:06:19.616 --> 00:06:21.426 A:middle And we first want to find a sub 1. 00:06:22.356 --> 00:06:27.366 A:middle We plug in 1 for n. And we have 1 over the quantity 1 plus 2. 00:06:27.586 --> 00:06:28.886 A:middle Can you cancel the ones? 00:06:29.086 --> 00:06:32.346 A:middle No. You need to do what's in the denominator first. 00:06:33.046 --> 00:06:34.386 A:middle 1 plus 2 is 3. 00:06:34.596 --> 00:06:35.686 A:middle So, we get one third. 00:06:36.116 --> 00:06:37.226 A:middle That does not reduce. 00:06:37.436 --> 00:06:39.176 A:middle So, a sub 1 is one third. 00:06:40.156 --> 00:06:43.056 A:middle Now, we want to find the second term, the second term 00:06:43.056 --> 00:06:47.036 A:middle of the sequence, we plug in 2 for n. We have 2 00:06:47.036 --> 00:06:48.846 A:middle over the quantity 2 plus 2. 00:06:49.546 --> 00:06:51.376 A:middle We do what's in the denominator first 00:06:51.376 --> 00:06:53.376 A:middle because of our order of operations. 00:06:53.626 --> 00:06:55.906 A:middle Order of operation says, do what's in the numerator, 00:06:56.246 --> 00:06:59.356 A:middle do what's in the denominator and then reduce if possible. 00:06:59.756 --> 00:07:03.746 A:middle So, we have to add the 2 plus 2, and we get 4. 00:07:04.506 --> 00:07:06.586 A:middle Now, can we reduce two 4s? 00:07:06.706 --> 00:07:09.886 A:middle Yes. Two 4s reduce to one half. 00:07:09.886 --> 00:07:12.406 A:middle [variable] a sub 2 is one half. 00:07:12.506 --> 00:07:14.786 A:middle [variable] a sub 3, we substitute in 3 for n 00:07:14.786 --> 00:07:19.386 A:middle in the denominator, we get 3 plus 2, which is 5. 00:07:19.786 --> 00:07:21.776 A:middle And three fifths does not reduce. 00:07:21.776 --> 00:07:23.256 A:middle [variable] a sub 3 is three fifths. 00:07:24.596 --> 00:07:27.716 A:middle [variable] a sub 4, we substitute in 4 for n. 00:07:28.566 --> 00:07:31.606 A:middle In the denominator, we get 4 plus 2, which is 6. 00:07:32.056 --> 00:07:34.006 A:middle And we have 4 over 6. 00:07:34.676 --> 00:07:38.076 A:middle Four over 6 does reduce by dividing the numerator 00:07:38.076 --> 00:07:41.416 A:middle and denominator by 2, and we get two thirds, 00:07:41.786 --> 00:07:43.266 A:middle a sub 4 is two thirds. 00:07:43.896 --> 00:07:48.766 A:middle Therefore the first four terms are one third, one half, 00:07:49.666 --> 00:07:51.806 A:middle three fifths and two thirds. 00:07:59.046 --> 00:08:00.826 A:middle Pause the video and try these problems. 00:08:01.516 --> 00:08:11.676 A:middle [ Pause ] 00:08:12.176 --> 00:08:15.106 A:middle Objective two, successive differences. 00:08:16.596 --> 00:08:19.176 A:middle We can use what are called successive differences 00:08:19.226 --> 00:08:21.466 A:middle to find the pattern for some sequences. 00:08:22.666 --> 00:08:26.956 A:middle In order to do this, we find the difference between each pair 00:08:26.956 --> 00:08:28.816 A:middle of consecutive terms in the sequence. 00:08:29.826 --> 00:08:32.556 A:middle Then we find the difference between each pair 00:08:32.976 --> 00:08:34.826 A:middle of the original differences. 00:08:35.366 --> 00:08:37.096 A:middle We continue this process 00:08:37.096 --> 00:08:39.896 A:middle until we discover a pattern in the differences. 00:08:41.026 --> 00:08:43.826 A:middle We can use this pattern to determine the next term. 00:08:45.076 --> 00:08:46.176 A:middle Let me show an example. 00:08:47.646 --> 00:08:49.386 A:middle Use successive differences 00:08:49.986 --> 00:08:51.946 A:middle to determine the next term in the sequence. 00:08:52.566 --> 00:08:58.856 A:middle Our sequences 3, 5, 8, 12, 17. 00:08:59.926 --> 00:09:00.846 A:middle Here's our sequence. 00:09:01.966 --> 00:09:04.296 A:middle So, difference means to subtract. 00:09:05.186 --> 00:09:07.526 A:middle So, 5 minus 3 is 2. 00:09:08.446 --> 00:09:10.106 A:middle 8 minus 5 is 3. 00:09:11.886 --> 00:09:14.246 A:middle 12 minus 8 is 4. 00:09:15.976 --> 00:09:18.466 A:middle 17 minus 12 is 5. 00:09:18.516 --> 00:09:20.246 A:middle So, we found the difference 00:09:20.296 --> 00:09:25.086 A:middle between each successive terms in the sequence. 00:09:25.466 --> 00:09:26.946 A:middle Now, we're going to do this again. 00:09:27.966 --> 00:09:30.876 A:middle Find the difference between those differences. 00:09:31.926 --> 00:09:34.136 A:middle 3 minus 2 is 1. 00:09:35.536 --> 00:09:37.716 A:middle 4 minus 3 is 1. 00:09:38.736 --> 00:09:40.996 A:middle 5 minus 4 is 1. 00:09:41.386 --> 00:09:42.676 A:middle Bingo, we have a pattern. 00:09:43.246 --> 00:09:44.616 A:middle Okay, so now we have a pattern. 00:09:44.776 --> 00:09:46.406 A:middle We know it's going to be a 1. 00:09:47.746 --> 00:09:50.066 A:middle All right, so we know that the next number 00:09:50.066 --> 00:09:51.446 A:middle down the bottom is going to be a 1. 00:09:52.416 --> 00:09:54.906 A:middle So, how do we figure out the next number here? 00:09:55.716 --> 00:09:56.396 A:middle Well, let's see. 00:09:56.856 --> 00:09:58.716 A:middle 5 minus 4 is 1. 00:10:00.056 --> 00:10:01.976 A:middle So, you would subtract to get the number down here. 00:10:02.166 --> 00:10:05.356 A:middle So, what do you do to figure out what this number is here? 00:10:05.986 --> 00:10:07.516 A:middle You would add, wouldn't you? 00:10:07.516 --> 00:10:10.556 A:middle Yes. 4 plus 1 is five. 00:10:11.126 --> 00:10:13.856 A:middle 3 plus 1 is 4, exactly. 00:10:13.856 --> 00:10:15.126 A:middle So, we're going to add to figure 00:10:15.126 --> 00:10:16.826 A:middle out what number goes right there, aren't we? 00:10:17.266 --> 00:10:18.226 A:middle Yes, we're going to add. 00:10:18.706 --> 00:10:21.456 A:middle So, 5 plus 1 is 6. 00:10:21.456 --> 00:10:23.896 A:middle Now, we do the same thing for the number that's missing 00:10:23.896 --> 00:10:25.696 A:middle up here, which is our answer to our problem. 00:10:26.516 --> 00:10:30.516 A:middle 17 plus 6 is 23. 00:10:31.296 --> 00:10:33.936 A:middle The next term in our sequence is 23. 00:10:41.046 --> 00:10:42.436 A:middle Use excessive differences 00:10:42.486 --> 00:10:44.386 A:middle to determine the next term in the sequence. 00:10:44.386 --> 00:10:50.736 A:middle And our sequence here is 2, 9, 28, 65, 126. 00:10:52.916 --> 00:10:56.576 A:middle So, we subtract, 9 minus 2 is 7. 00:10:57.936 --> 00:11:01.326 A:middle 28 minus 9 is 19. 00:11:03.476 --> 00:11:07.606 A:middle 65 minus 28 is 37. 00:11:08.966 --> 00:11:14.446 A:middle 126 minus 65 is 61. 00:11:15.596 --> 00:11:16.676 A:middle And we don't see a pattern yet. 00:11:16.676 --> 00:11:17.686 A:middle At least, I don't. 00:11:19.446 --> 00:11:21.156 A:middle All right, next line. 00:11:22.656 --> 00:11:25.246 A:middle 19 minus 7 is 12. 00:11:27.136 --> 00:11:30.776 A:middle 37 minus 19 is 18. 00:11:32.726 --> 00:11:37.096 A:middle 61 minus 37 is 24. 00:11:37.096 --> 00:11:40.126 A:middle Ah, those all different by 6. 00:11:40.646 --> 00:11:46.616 A:middle Yeah, 18 minus 12 is 6, and 24 minus 18 is 6. 00:11:47.646 --> 00:11:48.486 A:middle All right. 00:11:50.036 --> 00:11:54.256 A:middle Let's put up the next grid, we know that the next -- 00:11:54.716 --> 00:11:56.946 A:middle on the next line, we will have a 6. 00:11:58.296 --> 00:12:02.566 A:middle So, we add 24 plus 6 and get 30. 00:12:02.566 --> 00:12:03.696 A:middle Thirty goes here. 00:12:05.126 --> 00:12:09.636 A:middle We add 61 plus 30 to get 91 here. 00:12:10.906 --> 00:12:13.076 A:middle And then to get our answer, which is the next term 00:12:13.076 --> 00:12:20.496 A:middle of our sequence, we had 126 plus 91 to get 217. 00:12:21.006 --> 00:12:23.976 A:middle The next term of our sequence is 217. 00:12:27.456 --> 00:12:28.976 A:middle Pause the video and try these problems. 00:12:29.516 --> 00:12:43.026 A:middle [ Pause ] 00:12:43.526 --> 00:12:46.616 A:middle Objective three, arithmetic sequences. 00:12:48.126 --> 00:12:50.266 A:middle In this example, we're going to again, 00:12:50.316 --> 00:12:52.016 A:middle use successive differences 00:12:52.046 --> 00:12:53.866 A:middle to determine the next term in that sequence. 00:12:54.736 --> 00:13:00.126 A:middle Sequence is 9, 14, 19, 24, 29. 00:13:00.996 --> 00:13:04.356 A:middle We subtract 14 minus 9 is 5. 00:13:05.516 --> 00:13:07.076 A:middle 19 minus 14 is 5. 00:13:07.576 --> 00:13:09.056 A:middle 24 minus 19 is 5. 00:13:09.746 --> 00:13:12.316 A:middle 29 minus 24 is 5. 00:13:12.316 --> 00:13:14.256 A:middle And here we quickly determine a pattern. 00:13:15.146 --> 00:13:18.336 A:middle We add 5 to each term to get the next term. 00:13:18.726 --> 00:13:21.066 A:middle So, the next term in our sequence is 34. 00:13:22.466 --> 00:13:25.456 A:middle Any sequence where there is a common difference 00:13:25.456 --> 00:13:29.606 A:middle between successive terms, that's called an arithmetic sequence. 00:13:30.276 --> 00:13:33.866 A:middle We're going to use d to denote the common difference. 00:13:34.126 --> 00:13:37.396 A:middle So, in the above example, d is five. 00:13:39.676 --> 00:13:42.286 A:middle Let's determine a formula for this sequence. 00:13:43.046 --> 00:13:45.126 A:middle So, the first term, a sub 1 is 9. 00:13:46.166 --> 00:13:47.966 A:middle [variable] a sub 2 is 14. 00:13:49.056 --> 00:13:50.676 A:middle [variable] a sub 3 is 19. 00:13:51.346 --> 00:13:53.226 A:middle [variable] a sub 4 is 24. 00:13:54.016 --> 00:13:56.026 A:middle And, a sub 5 is 29. 00:13:57.966 --> 00:13:59.596 A:middle How did we get to a sub 2? 00:14:00.696 --> 00:14:02.346 A:middle Well, we took the first term, 00:14:02.346 --> 00:14:05.016 A:middle which was 9, and we added 5 to it. 00:14:06.386 --> 00:14:09.236 A:middle To get the third term, we added another 5. 00:14:10.346 --> 00:14:12.956 A:middle To get the fourth term, we added another 5. 00:14:13.626 --> 00:14:16.246 A:middle To get to the fifth term, we added another 5. 00:14:16.856 --> 00:14:19.156 A:middle When you're looking for patterns, 00:14:19.466 --> 00:14:22.156 A:middle don't look at the beginning, look at the end. 00:14:22.806 --> 00:14:29.246 A:middle So, this fifth term here, we have a 9, and we have four 5s. 00:14:30.616 --> 00:14:35.426 A:middle The fourth term, we have the 9 plus three 5s. 00:14:36.726 --> 00:14:41.226 A:middle The third term, we have the 9 plus two 5s. 00:14:42.706 --> 00:14:45.476 A:middle Now, we want this to be a pattern. 00:14:45.686 --> 00:14:48.926 A:middle So, we want to make the first and the second terms 00:14:48.966 --> 00:14:50.966 A:middle to be the same pattern. 00:14:51.456 --> 00:14:54.036 A:middle So, even though it kind of looks dumb to write it, 00:14:54.226 --> 00:14:55.256 A:middle we're going to write it anyhow. 00:14:56.046 --> 00:14:59.516 A:middle So, this is 9 plus one 5s. 00:15:00.466 --> 00:15:04.096 A:middle And this is 9 plus no 5s. 00:15:04.096 --> 00:15:05.966 A:middle So, 9 plus zero 5s. 00:15:06.156 --> 00:15:09.146 A:middle Good. So, we've written down the pattern that we have here. 00:15:10.136 --> 00:15:12.096 A:middle So, we have 9 plus zero 5s. 00:15:12.096 --> 00:15:14.156 A:middle 9 plus one 5s and so on and so forth. 00:15:14.756 --> 00:15:17.616 A:middle Now, we're going to write that into a formula. 00:15:18.136 --> 00:15:21.316 A:middle So, we want a formula for a sub n, 00:15:21.316 --> 00:15:23.466 A:middle the general term in the sequence. 00:15:23.986 --> 00:15:24.646 A:middle What would we have? 00:15:25.336 --> 00:15:26.916 A:middle [variable] a sub n would be 9. 00:15:26.916 --> 00:15:32.076 A:middle Of course, there's always 9 plus something times 5. 00:15:32.486 --> 00:15:34.986 A:middle Well, let's see, what have we got here? 00:15:35.576 --> 00:15:38.306 A:middle When this is a 1, this is a zero. 00:15:38.306 --> 00:15:40.956 A:middle When this is a 2, this is a 1. 00:15:41.396 --> 00:15:44.446 A:middle When this is a 3, when the subscript is a 3, 00:15:44.576 --> 00:15:46.446 A:middle the number over here is a 2. 00:15:46.446 --> 00:15:48.766 A:middle It's always one smaller, isn't it? 00:15:49.236 --> 00:15:53.976 A:middle Yeah. The number of 5s is always one smaller than the subscript. 00:15:54.926 --> 00:16:00.326 A:middle So, when the subscript is a 5, then the number of 5s is a four. 00:16:01.236 --> 00:16:07.386 A:middle When the subscript is an n, then the number of 5s is n minus 1. 00:16:08.256 --> 00:16:13.086 A:middle So, the formula for the sequence is a sub n equals 9 plus the 00:16:13.086 --> 00:16:15.966 A:middle quantity n minus 1 times 5. 00:16:17.966 --> 00:16:20.856 A:middle Nine was our first term, a sub 1. 00:16:21.636 --> 00:16:23.686 A:middle Five was our common difference. 00:16:23.686 --> 00:16:27.106 A:middle And the common difference is normally denoted, d. So, 00:16:27.106 --> 00:16:31.626 A:middle if we replace 9, which is a sub 1, and 5 with d, 00:16:32.416 --> 00:16:36.216 A:middle we get the formula in general for an arithmetic sequence. 00:16:36.766 --> 00:16:41.876 A:middle [variable] a sub n equals a sub 1 plus the quantity n minus 1 00:16:41.876 --> 00:16:45.966 A:middle times d. This formula can be used to find terms 00:16:45.966 --> 00:16:46.946 A:middle in an arithmetic sequence. 00:16:54.046 --> 00:16:57.496 A:middle Example, write a formula for the nth term of the sequence 00:16:57.936 --> 00:17:00.156 A:middle and then use it to find the 10th term. 00:17:01.136 --> 00:17:08.276 A:middle Our sequence is negative 7, negative 4, negative 1, 2, 5, 00:17:08.426 --> 00:17:09.326 A:middle and so on and so forth. 00:17:11.166 --> 00:17:14.016 A:middle Negative 4 minus negative 7, 00:17:14.016 --> 00:17:17.686 A:middle that would be negative 4 plus 7, which gives us 3. 00:17:17.686 --> 00:17:18.966 A:middle So, the difference is three. 00:17:20.066 --> 00:17:24.916 A:middle Negative 1 minus negative 4 would be negative 1 plus 4. 00:17:25.146 --> 00:17:27.096 A:middle And that difference is also 3. 00:17:28.346 --> 00:17:32.346 A:middle So, we right now know that this is an arithmetic sequence 00:17:32.546 --> 00:17:34.436 A:middle where the common difference is 3. 00:17:34.696 --> 00:17:37.776 A:middle Let's go ahead and do it for the other 2 terms. 00:17:38.016 --> 00:17:39.966 A:middle But we already know that's arithmetic 00:17:39.966 --> 00:17:41.226 A:middle since we've checked it twice. 00:17:41.736 --> 00:17:43.906 A:middle 2 minus negative 1 is also 3. 00:17:43.906 --> 00:17:45.256 A:middle And 5 minus 2 is three. 00:17:46.766 --> 00:17:48.856 A:middle You don't have to do the successive differences 00:17:49.266 --> 00:17:50.846 A:middle if you've already checked it twice, 00:17:50.846 --> 00:17:52.936 A:middle you know that it's an arithmetic sequence. 00:17:53.216 --> 00:17:54.256 A:middle All right. 00:17:54.286 --> 00:17:55.896 A:middle So, we know it's an arithmetic sequence. 00:17:55.896 --> 00:17:57.276 A:middle We know the common difference is three. 00:17:57.276 --> 00:17:58.526 A:middle So, we can apply this formula. 00:17:59.586 --> 00:18:02.456 A:middle [variable] a sub 1 is negative 7, and d is 3. 00:18:02.996 --> 00:18:04.436 A:middle What are we looking for? 00:18:04.626 --> 00:18:05.236 A:middle A formula? 00:18:05.676 --> 00:18:08.896 A:middle We're looking for a formula for the nth term, all right? 00:18:08.896 --> 00:18:10.346 A:middle So, we start off with this formula. 00:18:11.146 --> 00:18:13.436 A:middle [variable] a sub 1 is negative 7, and d is three. 00:18:13.856 --> 00:18:17.326 A:middle So, we substitute in negative 7 for a sub one. 00:18:17.726 --> 00:18:21.686 A:middle We substitute in 3 for d. And we should simplify our formula. 00:18:22.446 --> 00:18:23.746 A:middle We distribute the 3. 00:18:24.136 --> 00:18:28.606 A:middle So, we get negative 7 plus 3n minus 3. 00:18:29.616 --> 00:18:32.326 A:middle So, a sub n is equal to 3n minus 10. 00:18:32.326 --> 00:18:35.066 A:middle And that's the formula we were looking for, 00:18:35.436 --> 00:18:37.526 A:middle which is the first answer for our problem. 00:18:39.126 --> 00:18:42.306 A:middle The second question was use it to find the 10th term. 00:18:43.336 --> 00:18:45.326 A:middle So, we're looking for a sub 10. 00:18:45.976 --> 00:18:50.506 A:middle [variable] a sub 10 would be 3 times 10 minus 10, which is 20. 00:18:51.296 --> 00:18:53.856 A:middle The 10th term of the sequence is 20. 00:18:59.086 --> 00:19:00.726 A:middle Write a formula for the nth term of the sequence 00:19:00.726 --> 00:19:02.206 A:middle and use it to find the 12 term. 00:19:03.576 --> 00:19:08.196 A:middle The sequence is 20, 16, 12, 8, 4. 00:19:09.976 --> 00:19:10.706 A:middle We write it out. 00:19:11.576 --> 00:19:14.426 A:middle 16 minus 20 is negative 4. 00:19:15.736 --> 00:19:18.606 A:middle 12 minus 16 is negative 4. 00:19:18.776 --> 00:19:21.236 A:middle We know we have an arithmetic sequence. 00:19:21.636 --> 00:19:23.526 A:middle And d is negative 4. 00:19:23.696 --> 00:19:26.626 A:middle There's no reason to do the rest of the successive differences. 00:19:26.996 --> 00:19:30.256 A:middle We know we have an arithmetic sequence and d is negative four. 00:19:31.786 --> 00:19:33.396 A:middle Here's the formula that we use. 00:19:34.226 --> 00:19:37.336 A:middle We know that a sub 1 is 20, and d is negative 4. 00:19:37.336 --> 00:19:38.646 A:middle Let's substitute those in. 00:19:39.816 --> 00:19:46.526 A:middle So, a sub n is equal to 20 plus the quantity n minus 1 times -- 00:19:46.526 --> 00:19:49.286 A:middle now, be careful to put d in parentheses 00:19:49.286 --> 00:19:50.406 A:middle because d is negative. 00:19:52.236 --> 00:19:54.166 A:middle Now, we're going to distribute the negative 4. 00:19:55.026 --> 00:19:59.986 A:middle So, we get 20 minus 4n plus 4. 00:20:01.456 --> 00:20:05.996 A:middle So, we get a sub n is equal to 24 minus 4n. 00:20:06.636 --> 00:20:09.706 A:middle This is the formula that we were asked for in the problem. 00:20:10.046 --> 00:20:11.886 A:middle So the formula for the nth term 00:20:11.886 --> 00:20:17.076 A:middle of the sequence is a sub n equals 24 minus 4n. 00:20:17.406 --> 00:20:20.996 A:middle That's the first answer that we were asked for in the problem. 00:20:22.556 --> 00:20:24.736 A:middle Next, we were asked for the 12th term. 00:20:25.646 --> 00:20:28.046 A:middle To find the 12th term, we would substitute in n12 00:20:28.046 --> 00:20:33.946 A:middle for n. [variable] a sub 12 would be 24 minus 4 times 12. 00:20:33.946 --> 00:20:38.336 A:middle We'd put in 12 for n, and we get negative 24. 00:20:38.926 --> 00:20:42.976 A:middle The 12th term of the sequence is negative 24. 00:20:49.066 --> 00:20:51.876 A:middle Determine the number of terms in this sequence using the formula. 00:20:53.136 --> 00:20:57.116 A:middle 7, 10, 13 and so on and so forth to 46. 00:20:57.576 --> 00:21:00.586 A:middle Well, 10 minus 7 is 3. 00:21:01.696 --> 00:21:03.386 A:middle So, the common difference is 3, right? 00:21:03.386 --> 00:21:05.036 A:middle But, wait a minute, you have to check it twice, 00:21:05.036 --> 00:21:06.596 A:middle you always have to check it twice. 00:21:07.266 --> 00:21:09.546 A:middle 13 minus 10 is also 3. 00:21:09.576 --> 00:21:14.166 A:middle Okay, so we have an arithmetic sequence, and d is three, 00:21:14.776 --> 00:21:16.216 A:middle the common difference is 3. 00:21:16.816 --> 00:21:21.926 A:middle So, we can use this formula, a sub 1 is 7. 00:21:22.466 --> 00:21:23.396 A:middle And d is 3. 00:21:25.646 --> 00:21:30.456 A:middle Now, we're used to finding a particular term in the sequence. 00:21:31.186 --> 00:21:32.476 A:middle We're not doing that this time. 00:21:33.176 --> 00:21:35.256 A:middle The term is 46. 00:21:35.826 --> 00:21:38.086 A:middle We're trying to figure out which term it is. 00:21:38.306 --> 00:21:42.886 A:middle Now, we could sit there and go at 3, at 3, at 3, at 3, 00:21:42.886 --> 00:21:45.006 A:middle at 3 and figure out which term it is. 00:21:45.396 --> 00:21:49.436 A:middle But what if instead of 46, it was 4,600 blah, blah, blah? 00:21:50.206 --> 00:21:51.386 A:middle No, no, no, no, no. 00:21:51.806 --> 00:21:53.686 A:middle We're supposed to show the math, 00:21:53.766 --> 00:21:55.376 A:middle we're supposed to use the formula. 00:21:55.376 --> 00:21:56.596 A:middle We're not going to get any credit 00:21:56.596 --> 00:21:57.686 A:middle if we don't use the formula. 00:21:57.686 --> 00:21:59.686 A:middle All right, so here's the formula. 00:21:59.686 --> 00:22:03.736 A:middle And the nth term is 46. 00:22:03.816 --> 00:22:07.086 A:middle The goal is to figure out what n is. 00:22:08.256 --> 00:22:11.016 A:middle So, the nth term is 46. 00:22:11.506 --> 00:22:13.816 A:middle [variable] a sub one, the first term is 7, 00:22:14.316 --> 00:22:16.596 A:middle and the common difference is 3. 00:22:16.786 --> 00:22:18.956 A:middle Our goal is to solve this equation 00:22:18.956 --> 00:22:22.196 A:middle for n. Let's distribute the 3. 00:22:23.276 --> 00:22:25.236 A:middle So, we get 3n minus 3 there. 00:22:26.166 --> 00:22:29.266 A:middle So, on the right hand side, we have 3n plus 4. 00:22:30.576 --> 00:22:32.626 A:middle We subtract 4 from both sides. 00:22:32.946 --> 00:22:35.276 A:middle So, we have 42 equals 3n. 00:22:36.656 --> 00:22:42.726 A:middle Then we divide both sides by 3, and we get 14 equals n. So, 00:22:42.866 --> 00:22:47.586 A:middle 46 is the 14th term of the sequence. 00:22:48.646 --> 00:22:51.606 A:middle [variable] a sub 14 is equal to 46. 00:22:52.576 --> 00:22:55.746 A:middle So, this sequence has 14 terms. 00:23:01.046 --> 00:23:04.116 A:middle Determine the number of terms in the sequence using the formula 00:23:04.116 --> 00:23:08.746 A:middle and the sequence is 8, 3, negative 2, 00:23:09.116 --> 00:23:11.886 A:middle so on and so forth, to negative 77. 00:23:13.476 --> 00:23:17.936 A:middle So, we subtract 3 minus 8 is negative 5. 00:23:18.406 --> 00:23:19.726 A:middle We have to check it twice. 00:23:20.746 --> 00:23:24.566 A:middle Negative 2 minus 3 is negative 5. 00:23:25.346 --> 00:23:27.646 A:middle So, we have an arithmetic sequence. 00:23:28.136 --> 00:23:31.536 A:middle And the common difference, d, is negative 5. 00:23:32.786 --> 00:23:33.856 A:middle Here's our formula. 00:23:35.756 --> 00:23:39.426 A:middle [variable] a sub 1 is eight, and d is negative five. 00:23:40.026 --> 00:23:43.786 A:middle The last term of our sequence is negative 77. 00:23:44.186 --> 00:23:47.276 A:middle We want to find out how many terms are in this sequence. 00:23:47.536 --> 00:23:53.606 A:middle So, we want to find out which number term is negative 77. 00:23:54.026 --> 00:23:57.176 A:middle So, a sub what is equal to negative 77? 00:23:58.466 --> 00:24:02.796 A:middle So, we substitute in negative 77 for a sub n. And we want 00:24:02.796 --> 00:24:04.346 A:middle to find out what n is. 00:24:04.626 --> 00:24:06.306 A:middle How many terms are in the sequence, 00:24:06.306 --> 00:24:07.716 A:middle which term number is it? 00:24:08.186 --> 00:24:12.226 A:middle So, a sub 1 is 8, and d is negative 5. 00:24:12.226 --> 00:24:15.626 A:middle Be sure to put negative 5 in parenthesis since it's negative. 00:24:16.936 --> 00:24:19.626 A:middle On the next line, we distribute the negative 5. 00:24:20.126 --> 00:24:22.576 A:middle So, we get negative 5n plus five. 00:24:23.876 --> 00:24:26.326 A:middle Now, 8 plus 5 gives us 13. 00:24:26.326 --> 00:24:31.106 A:middle So, we have negative 77 equals 13 minus 5n. 00:24:31.876 --> 00:24:33.716 A:middle We subtract 13 from both sides 00:24:33.796 --> 00:24:37.106 A:middle to get negative 90 equals negative 5n. 00:24:38.366 --> 00:24:44.346 A:middle Divide both sides by negative 5, and we get 18 equals n. So, 00:24:44.346 --> 00:24:48.976 A:middle negative 77 is the 18th term of this sequence. 00:24:49.606 --> 00:24:51.976 A:middle So, the sequence has 18 terms. 00:24:58.046 --> 00:25:01.126 A:middle The first row in a stadium has 20 seats. 00:25:01.416 --> 00:25:03.786 A:middle The second row has 24 seats. 00:25:04.556 --> 00:25:06.896 A:middle The third row has 28 seats. 00:25:07.886 --> 00:25:10.586 A:middle This pattern continues to the back of the stadium. 00:25:11.746 --> 00:25:17.626 A:middle The stadium has 52 rows, how many seats are in the 25th row? 00:25:20.146 --> 00:25:26.386 A:middle So, the sequence is 20, 24, 28, and so on and so forth. 00:25:27.506 --> 00:25:31.316 A:middle [variable] a sub 1, the first term in the sequence is 20. 00:25:31.696 --> 00:25:33.736 A:middle [variable] a sub 2 is 24. 00:25:34.096 --> 00:25:35.966 A:middle [variable] a sub 3 is 28. 00:25:36.936 --> 00:25:37.876 A:middle What are we looking for? 00:25:40.256 --> 00:25:45.376 A:middle We're looking for how many seats are in the 25th row, 00:25:45.376 --> 00:25:48.386 A:middle which would be, a sub 25. 00:25:50.046 --> 00:25:52.036 A:middle Is this a particular type of sequence? 00:25:52.876 --> 00:25:54.406 A:middle Yes, it is arithmetic. 00:25:54.966 --> 00:25:57.536 A:middle 24 minus 20 is 4. 00:25:58.086 --> 00:26:01.466 A:middle Also, 28 minus 24 is also 4. 00:26:01.976 --> 00:26:03.776 A:middle This is an arithmetic sequence 00:26:03.776 --> 00:26:05.886 A:middle that has a common difference of 4. 00:26:06.876 --> 00:26:10.016 A:middle So, a sub one is 20 and d is 4. 00:26:11.236 --> 00:26:12.256 A:middle Here's our formula. 00:26:13.126 --> 00:26:16.476 A:middle And we're looking for a sub 25. 00:26:16.976 --> 00:26:19.776 A:middle So, we substitute in 20 for a sub 1. 00:26:21.106 --> 00:26:27.126 A:middle We substitute in 4 for d. And we also substitute in 25 for n, 00:26:27.126 --> 00:26:30.516 A:middle since we're looking for the 25th term in the sequence. 00:26:31.006 --> 00:26:33.196 A:middle We do what's in the parenthesis first. 00:26:33.766 --> 00:26:36.276 A:middle 25 minus 1 is 24. 00:26:37.306 --> 00:26:40.926 A:middle We multiply next, 24 times 4, 00:26:41.356 --> 00:26:44.976 A:middle and then we add the 20, to get 116. 00:26:45.696 --> 00:26:49.876 A:middle We were looking for a sub 25 and we got 116. 00:26:50.816 --> 00:26:55.976 A:middle There are 116 seats in the 25th row of the stadium. 00:27:00.046 --> 00:27:00.976 A:middle Pause the video and try these problems. 00:27:01.516 --> 00:27:14.546 A:middle [ Pause ] 00:27:15.046 --> 00:27:17.376 A:middle Objective four, arithmetic series. 00:27:18.936 --> 00:27:22.586 A:middle A series is the sum of the terms of a sequence. 00:27:23.556 --> 00:27:26.736 A:middle So, for a series you add up the terms of a sequence. 00:27:28.126 --> 00:27:31.576 A:middle The German mathematician, Carl Gauss at the age of 10, 00:27:31.986 --> 00:27:35.646 A:middle was asked to add up the first 100 natural numbers. 00:27:35.826 --> 00:27:37.166 A:middle Now, what's a natural number? 00:27:37.506 --> 00:27:39.966 A:middle Those are the counting numbers; 1, 2, 3, 00:27:39.966 --> 00:27:41.066 A:middle 4 and so on and so forth. 00:27:41.296 --> 00:27:43.706 A:middle So, he was asked to add up these numbers here. 00:27:44.576 --> 00:27:46.536 A:middle He noticed the following pattern. 00:27:47.296 --> 00:27:51.056 A:middle If you take the first number 1, and the last number 100 00:27:51.056 --> 00:27:52.566 A:middle and add them up, you get 101. 00:27:53.186 --> 00:27:56.316 A:middle And if you take the next number, 2 and the next 00:27:56.316 --> 00:28:00.446 A:middle to last number, 99 you also get 101. 00:28:00.446 --> 00:28:05.426 A:middle In fact, if you take 3 and 98, you also get one 101. 00:28:05.426 --> 00:28:08.746 A:middle And if you take 4 and 97, you also get 101. 00:28:09.176 --> 00:28:12.156 A:middle And if you take 5 and 96, you also get 101. 00:28:12.156 --> 00:28:15.936 A:middle And in fact, every single pairing like that, you're going 00:28:15.936 --> 00:28:17.616 A:middle to get 101 every single time. 00:28:19.356 --> 00:28:21.556 A:middle How many pairs are like that? 00:28:22.066 --> 00:28:23.946 A:middle Every single pair is going to be like that. 00:28:23.946 --> 00:28:25.666 A:middle In fact, how many pairs are there? 00:28:26.466 --> 00:28:27.856 A:middle There's going to be 50 of those pairs, 00:28:27.856 --> 00:28:29.256 A:middle since there's 100 numbers, aren't there? 00:28:29.776 --> 00:28:31.236 A:middle There's going to be 50 of those pairs, 00:28:31.236 --> 00:28:33.866 A:middle and every single pair is going to add up to 101. 00:28:34.666 --> 00:28:35.536 A:middle So, what's the sum? 00:28:36.056 --> 00:28:41.536 A:middle The sum is 50 times 101 which turns out to be 5,050. 00:28:42.976 --> 00:28:45.866 A:middle Kind of a cool thing for a 10 year old to do. 00:28:46.136 --> 00:28:48.016 A:middle We now want to find a formula for adding 00:28:48.016 --> 00:28:49.726 A:middle up a certain number of natural numbers. 00:28:50.816 --> 00:28:56.566 A:middle So, we are going to add up 1 plus 2 plus 3, plus 4, 00:28:56.566 --> 00:28:57.866 A:middle and so on and so forth. 00:28:58.366 --> 00:29:00.566 A:middle The last number is just going to be n, it's not going 00:29:00.566 --> 00:29:04.776 A:middle to be 100 it's going to be just some specific natural number. 00:29:04.776 --> 00:29:06.076 A:middle We don't know what it is. 00:29:06.596 --> 00:29:09.056 A:middle Now, what would the number right before n be? 00:29:09.696 --> 00:29:11.216 A:middle Well, since it's a natural number, 00:29:11.216 --> 00:29:13.806 A:middle the number right before it would be 1 smaller than it. 00:29:13.846 --> 00:29:15.206 A:middle So, that the n minus 1. 00:29:16.786 --> 00:29:20.926 A:middle And the number right before that would be n minus 2, yeah? 00:29:21.076 --> 00:29:27.386 A:middle Okay. Now, we're going to take that whole sum, and we're going 00:29:27.386 --> 00:29:28.366 A:middle to write it backwards. 00:29:29.746 --> 00:29:33.556 A:middle So, instead of from the left, going 1, 2, 3. 00:29:33.896 --> 00:29:37.286 A:middle Now, from the right, we're going to go 1 plus 2 plus 3 00:29:37.286 --> 00:29:38.376 A:middle and so on and so forth. 00:29:38.436 --> 00:29:40.976 A:middle And then we're going to have the n minus 2, and then we're going 00:29:40.976 --> 00:29:43.016 A:middle to have the n minus 1, and then we're going to have the n. 00:29:43.516 --> 00:29:46.736 A:middle And then the left hand side, we still have some equals. 00:29:47.746 --> 00:29:50.756 A:middle Now, we're going to add up these two sums. 00:29:51.096 --> 00:29:52.436 A:middle So, on the left hand side, 00:29:52.796 --> 00:29:57.046 A:middle we have sum plus sum equals twice the sum, okay? 00:29:57.966 --> 00:29:59.746 A:middle So, we're going to add up these two equations. 00:30:01.216 --> 00:30:02.536 A:middle Now, what do we get when we add up these? 00:30:03.106 --> 00:30:06.206 A:middle 1 plus n is just n plus 1. 00:30:06.206 --> 00:30:07.386 A:middle We can write it in any order. 00:30:07.386 --> 00:30:09.506 A:middle 1 plus n or n plus 1 doesn't make a difference. 00:30:10.536 --> 00:30:13.326 A:middle What do we get when we add 2 plus n minus 1? 00:30:13.746 --> 00:30:15.526 A:middle Well, 2 minus 1 is 1. 00:30:15.806 --> 00:30:17.506 A:middle So, I would get n plus 1. 00:30:18.446 --> 00:30:19.926 A:middle Oh, that's the same thing, cool. 00:30:20.926 --> 00:30:23.686 A:middle What do you get when you add 3 and n minus 2? 00:30:23.976 --> 00:30:26.366 A:middle Well, 3 minus 2 is positive 1. 00:30:26.466 --> 00:30:27.986 A:middle So, I get n plus 1. 00:30:28.206 --> 00:30:30.796 A:middle Huh, the parentheses are all the same. 00:30:31.536 --> 00:30:32.426 A:middle Then, let's come down here. 00:30:33.346 --> 00:30:36.596 A:middle [variable] n minus 2 and 3, I get the same thing; n plus 1. 00:30:37.416 --> 00:30:41.346 A:middle And n minus 1 and 2, when I add those up, I get n plus1. 00:30:42.106 --> 00:30:43.846 A:middle [variable] n plus 1 is n plus 1. 00:30:44.156 --> 00:30:46.886 A:middle Wow, I get the same thing 00:30:46.886 --> 00:30:48.256 A:middle in every single one of the parentheses. 00:30:48.566 --> 00:30:51.866 A:middle So, when I add up these two things on the right hand side, 00:30:51.866 --> 00:30:53.926 A:middle every single time I get n plus 1. 00:30:54.946 --> 00:30:57.296 A:middle And I have the little dot, dot, dot in the middle showing 00:30:57.296 --> 00:30:59.776 A:middle that I get the same thing every single time all the way 00:30:59.776 --> 00:31:00.296 A:middle across the board. 00:31:01.076 --> 00:31:04.086 A:middle Now, how many sets of parentheses are there? 00:31:04.086 --> 00:31:05.816 A:middle You're going to say 6, right? 00:31:05.816 --> 00:31:08.096 A:middle No, no, no, no, no, there's a little dot dot dot in the middle 00:31:08.096 --> 00:31:10.516 A:middle that says there's a whole bunch of [inaudible] in between 00:31:11.506 --> 00:31:12.786 A:middle that we're not seeing, right? 00:31:12.786 --> 00:31:13.676 A:middle We can't write them all. 00:31:14.866 --> 00:31:16.266 A:middle How many of them are there? 00:31:16.946 --> 00:31:20.856 A:middle Well, let's look them up top; 1, 2, 3. 00:31:20.856 --> 00:31:25.426 A:middle And then so on and so forth to n. There were n numbers 00:31:25.426 --> 00:31:26.736 A:middle across here, weren't there? 00:31:27.126 --> 00:31:29.496 A:middle Yes, there were n natural numbers 00:31:29.496 --> 00:31:30.716 A:middle that we're trying to add up. 00:31:31.226 --> 00:31:34.616 A:middle So, there would be n sets of parentheses down here. 00:31:36.676 --> 00:31:40.506 A:middle So, on the right hand side, we have n times 00:31:40.906 --> 00:31:43.036 A:middle that parenthesis, n plus 1. 00:31:45.306 --> 00:31:48.606 A:middle So, our equation is twice the sum is equal 00:31:48.606 --> 00:31:51.286 A:middle to n times the parenthesis n plus 1. 00:31:52.236 --> 00:31:54.866 A:middle If we divide both sides by two, we have our formula. 00:31:55.716 --> 00:31:59.846 A:middle The sum of the first n natural numbers is equal to, 00:32:00.856 --> 00:32:03.976 A:middle n times the quantity n plus one all over 2. 00:32:04.566 --> 00:32:06.906 A:middle And there's our form of that we're looking for. 00:32:08.296 --> 00:32:12.106 A:middle Now, since we have Carl Gauss's example, 00:32:12.436 --> 00:32:13.946 A:middle let's go ahead and check it out. 00:32:14.156 --> 00:32:17.036 A:middle He was adding up the first 100 natural numbers. 00:32:17.036 --> 00:32:19.766 A:middle So, let's put in 100 for n. So, 00:32:19.766 --> 00:32:24.526 A:middle we have 100 times the quantity, 100 plus 1 over 2. 00:32:25.156 --> 00:32:29.516 A:middle So, that's 100 times 101 all over 2. 00:32:30.316 --> 00:32:34.356 A:middle And I don't need to, but I could take that 100 over 2, 00:32:34.356 --> 00:32:36.076 A:middle and pull that over to the side. 00:32:36.076 --> 00:32:39.216 A:middle 100 divided by 2 is 50. 00:32:39.256 --> 00:32:44.286 A:middle Yeah, there were 50 of those 101 pairings, weren't there? 00:32:45.216 --> 00:32:47.976 A:middle And that gives me the exact same answer 5,050. 00:32:56.436 --> 00:32:59.676 A:middle Determine the sum of the first 500 natural numbers. 00:33:00.076 --> 00:33:02.336 A:middle And we are, of course, supposed to use a formula here. 00:33:02.866 --> 00:33:05.436 A:middle So, here's the formula that we just developed. 00:33:07.226 --> 00:33:08.966 A:middle So, n is equal to 500. 00:33:09.706 --> 00:33:13.896 A:middle So, we have 500 times the quantity 500 plus 1 all over 2. 00:33:14.576 --> 00:33:18.426 A:middle So, we have 500 times 501, all divided by 2. 00:33:18.676 --> 00:33:22.406 A:middle And we get 125,250. 00:33:23.306 --> 00:33:29.696 A:middle The sum of the first 500 natural numbers is 125,250. 00:33:35.046 --> 00:33:39.396 A:middle Determine the following sum 42 plus 43 plus 44, 00:33:39.396 --> 00:33:41.276 A:middle and so on and so forth to 95. 00:33:42.656 --> 00:33:44.846 A:middle Well, this doesn't match the pattern we had before. 00:33:44.846 --> 00:33:47.006 A:middle Because before, we had 1 plus 2, plus 3. 00:33:47.006 --> 00:33:49.496 A:middle So, we have to make it match that pattern to be able to work 00:33:49.496 --> 00:33:51.816 A:middle with this, because that's the only formula we have. 00:33:52.386 --> 00:33:54.566 A:middle So, we need 1 plus 2 plus 3. 00:33:55.816 --> 00:33:58.056 A:middle So, 1 -- let's see. 00:33:58.416 --> 00:34:02.346 A:middle Well, 42 is 41 plus 1. 00:34:03.816 --> 00:34:08.726 A:middle And 43 is 41 plus 2. 00:34:08.726 --> 00:34:12.946 A:middle Yeah, I want to make this look like 1 plus 2 plus 3. 00:34:14.216 --> 00:34:18.076 A:middle So, if I wrote 42 as 41 plus. 00:34:19.176 --> 00:34:23.086 A:middle And 43 as 41 plus 2. 00:34:23.726 --> 00:34:31.246 A:middle And 44 would be 41 plus 3, so on and so forth. 00:34:31.676 --> 00:34:37.406 A:middle Now, I simply subtract 41 from 95 and get 54. 00:34:38.856 --> 00:34:41.276 A:middle So, I have it written out. 00:34:41.626 --> 00:34:42.616 A:middle Now, what am I going to do? 00:34:42.746 --> 00:34:45.226 A:middle I'm going to take out all those 41s. 00:34:45.926 --> 00:34:47.946 A:middle How many 41s are there? 00:34:48.346 --> 00:34:50.776 A:middle One, two, three, and so on and so forth. 00:34:50.776 --> 00:34:52.386 A:middle There are 54 of those, aren't there? 00:34:52.526 --> 00:34:54.346 A:middle There are 54 41s. 00:34:54.346 --> 00:34:56.756 A:middle So, I'm going to take out 54 41s. 00:34:57.746 --> 00:35:03.286 A:middle So, I'm going to have 54 times 41 plus the sum, 1 plus 2, 00:35:03.286 --> 00:35:06.046 A:middle plus 3 and so on and so forth to 54. 00:35:06.136 --> 00:35:08.676 A:middle Oh, we know how to add that up because we have a formula. 00:35:09.376 --> 00:35:10.256 A:middle Here's our formula. 00:35:10.826 --> 00:35:13.786 A:middle And we're going to use that to add up the numbers 00:35:13.786 --> 00:35:14.916 A:middle that are in the parenthesis. 00:35:15.436 --> 00:35:16.906 A:middle [variable] n is 54. 00:35:18.246 --> 00:35:21.776 A:middle So, we multiply 54 times 41 to get 2,214. 00:35:22.506 --> 00:35:25.786 A:middle We substitute in 54 for n. So, 00:35:25.786 --> 00:35:28.766 A:middle we have 54 times the quantity 54 plus 1 00:35:29.076 --> 00:35:30.896 A:middle on top and divide it by 2. 00:35:32.166 --> 00:35:36.036 A:middle So, we have 55 in the parenthesis, the numerator. 00:35:36.726 --> 00:35:38.136 A:middle We get 2,970. 00:35:39.176 --> 00:35:41.516 A:middle We divided to get 1,485. 00:35:42.466 --> 00:35:46.016 A:middle And finally, we add to get 3,699. 00:35:47.526 --> 00:35:50.476 A:middle Our sum is 3699. 00:35:55.046 --> 00:35:56.696 A:middle Pause the video and try these problems. 00:36:09.046 --> 00:36:11.836 A:middle Objective five, geometric sequences. 00:36:12.806 --> 00:36:19.306 A:middle Consider the sequence 3, 6, 12, 24, 48, and so on and so forth. 00:36:21.556 --> 00:36:23.726 A:middle Well, 6 minus 3 is 3. 00:36:23.726 --> 00:36:26.616 A:middle So, it's an arithmetic sequence where d is 3, right? 00:36:27.766 --> 00:36:30.896 A:middle Wait a minute, we've got to check it twice, don't we? 00:36:31.926 --> 00:36:35.146 A:middle 12 minus 6 is 6. 00:36:35.146 --> 00:36:37.406 A:middle So, this is not arithmetic. 00:36:38.206 --> 00:36:39.646 A:middle [variable] d is not 3. 00:36:39.646 --> 00:36:40.536 A:middle That doesn't work. 00:36:41.286 --> 00:36:45.696 A:middle And in fact, the pattern here is that we multiply each term by 2 00:36:45.696 --> 00:36:46.926 A:middle to get to the next term. 00:36:48.126 --> 00:36:51.186 A:middle Another way to put that is if you divide any term 00:36:51.186 --> 00:36:54.536 A:middle by the previous term, we get the same result. 00:36:55.676 --> 00:36:59.956 A:middle So, let's take 6 and divide it by the term that comes before 3. 00:37:00.496 --> 00:37:02.126 A:middle And that gives you 2. 00:37:03.126 --> 00:37:05.416 A:middle Let's take 12 and divide it by the term 00:37:05.416 --> 00:37:07.246 A:middle that comes before it, 6. 00:37:07.666 --> 00:37:09.186 A:middle And we get 2. 00:37:09.186 --> 00:37:11.256 A:middle Ah, we checked it twice, so we're good. 00:37:11.746 --> 00:37:14.326 A:middle So, this is a geometric sequence. 00:37:14.706 --> 00:37:19.666 A:middle And the common ratio, which we call 4 r is 2. 00:37:20.676 --> 00:37:22.476 A:middle Let's check it a third time just for fun. 00:37:22.916 --> 00:37:28.166 A:middle 24 divided by the term right before it, 12 is 2, again. 00:37:29.026 --> 00:37:33.406 A:middle A sequence that has a common ratio is called a geometric 00:37:33.406 --> 00:37:37.546 A:middle sequence, and we call the common ratio, r. So, 00:37:37.546 --> 00:37:40.306 A:middle this is a geometric sequence. 00:37:40.476 --> 00:37:43.636 A:middle And the common ratio for this sequence is 2. 00:37:49.046 --> 00:37:51.006 A:middle Determine the next term in the sequence. 00:37:51.466 --> 00:37:56.296 A:middle And our sequence is 5, 15, 45 and so on and so forth. 00:37:57.966 --> 00:38:01.106 A:middle We notice that if we take 15 and divide it 00:38:01.106 --> 00:38:03.986 A:middle by the term before it, which is 5, we get 3. 00:38:04.546 --> 00:38:09.476 A:middle Also, if we take 45 and divide it by 15, we also get 3. 00:38:10.316 --> 00:38:13.166 A:middle Therefore, this is a geometric sequence 00:38:13.566 --> 00:38:16.196 A:middle with a common ratio r of 3. 00:38:17.786 --> 00:38:22.496 A:middle To find the next term then, we would multiply the last term, 00:38:22.496 --> 00:38:25.506 A:middle 45, by the common ratio 3. 00:38:26.396 --> 00:38:30.646 A:middle 45 times 3 is 135. 00:38:32.266 --> 00:38:36.286 A:middle The next term of the sequence will be 135 00:38:36.616 --> 00:38:39.246 A:middle because you multiply each term by three 00:38:39.496 --> 00:38:41.906 A:middle to get the next term in the sequence. 00:38:49.046 --> 00:38:51.136 A:middle Write a formula for the nth term of the sequence 00:38:51.136 --> 00:38:52.906 A:middle and use it to find the 7th term. 00:38:53.516 --> 00:38:57.646 A:middle Our sequence is 5, 10, 20, 40 and so on and so forth. 00:38:58.396 --> 00:39:02.356 A:middle If we take 10 and divide it by 5, we get 2. 00:39:03.186 --> 00:39:07.266 A:middle Likewise, if we take 20 and divide it by 10, we also get 2. 00:39:08.226 --> 00:39:09.426 A:middle We checked it twice. 00:39:09.696 --> 00:39:12.176 A:middle So, we know that this is a geometric sequence, 00:39:12.496 --> 00:39:14.706 A:middle and the common ratio is 2. 00:39:16.086 --> 00:39:20.456 A:middle We were asked this time for a formula for the nth term. 00:39:20.766 --> 00:39:22.796 A:middle So, we need to look for a pattern. 00:39:24.076 --> 00:39:26.536 A:middle The first term, a sub 1 is five. 00:39:27.326 --> 00:39:30.226 A:middle The second term, a sub 2 is 10. 00:39:30.416 --> 00:39:31.636 A:middle And how do we get 10? 00:39:32.006 --> 00:39:35.076 A:middle We take the initial term, 5, and multiply it by 2. 00:39:35.686 --> 00:39:39.836 A:middle To get the third term of 20, we take 00:39:39.836 --> 00:39:42.496 A:middle and multiply the 10 by another 2. 00:39:43.076 --> 00:39:46.196 A:middle So, we have 5 times 2, times another 2. 00:39:46.936 --> 00:39:50.916 A:middle To get the 40, we take the initial 5 and multiply it 00:39:50.916 --> 00:39:53.556 A:middle by 2 and 2 and 2 again. 00:39:55.206 --> 00:39:59.566 A:middle Recall that 3 to the fourth power means three times itself 00:39:59.636 --> 00:40:00.566 A:middle 4 times. 00:40:01.266 --> 00:40:04.996 A:middle [variable] r to the fourth then means r times r times r times r. 00:40:05.586 --> 00:40:07.106 A:middle We're looking for a pattern. 00:40:07.596 --> 00:40:13.206 A:middle Forty is 5 times 2 times 2 times 2. 00:40:13.666 --> 00:40:18.536 A:middle So, we can rewrite that as 5 times 2 to the third power. 00:40:19.706 --> 00:40:22.026 A:middle I'll leave it on the outside of the parenthesis 00:40:22.026 --> 00:40:24.076 A:middle because sometimes we'll end up with a number 00:40:24.076 --> 00:40:25.666 A:middle in the parentheses that's negative, 00:40:25.836 --> 00:40:28.126 A:middle and then you would need it outside the parenthesis, 00:40:28.126 --> 00:40:28.906 A:middle the exponent. 00:40:29.666 --> 00:40:34.666 A:middle So, we have 5 times 2 to the third power for a sub 4. 00:40:35.616 --> 00:40:38.336 A:middle [variable] a sub 3, the third term then, 00:40:38.666 --> 00:40:41.786 A:middle would be 5 times 2 to the second power. 00:40:41.786 --> 00:40:45.376 A:middle We have 2 times 2, or 2 to the second power. 00:40:46.036 --> 00:40:48.516 A:middle We want to make them all look the same. 00:40:48.876 --> 00:40:51.286 A:middle So, the second term could be written 00:40:51.286 --> 00:40:54.446 A:middle as 5 times 2 to the first power. 00:40:55.236 --> 00:40:57.936 A:middle And recall that anything 00:40:57.936 --> 00:41:01.926 A:middle to the zero power is 1 other than zero itself. 00:41:02.666 --> 00:41:06.826 A:middle So, I know it looks silly, but we could write 5 00:41:07.086 --> 00:41:10.626 A:middle as 5 times 2 to the zero power. 00:41:11.596 --> 00:41:14.506 A:middle We're trying to make it so that we can see a pattern. 00:41:16.066 --> 00:41:19.486 A:middle So, we have 5 times 2 to varying powers. 00:41:20.246 --> 00:41:22.436 A:middle What would a sub 8 be? 00:41:23.876 --> 00:41:27.216 A:middle Well, the exponent is one smaller than the subscript. 00:41:27.216 --> 00:41:30.166 A:middle So, that would be 5 times 2 to the 7th power. 00:41:31.306 --> 00:41:34.326 A:middle [variable] a sub 30 then would be 5 times 2 00:41:34.326 --> 00:41:37.186 A:middle to the 29th power, one smaller. 00:41:38.516 --> 00:41:43.266 A:middle So, here we go with our formula, a sub n, the general term. 00:41:44.086 --> 00:41:48.716 A:middle The general term would be 5 times 2 to the power 00:41:48.716 --> 00:41:53.346 A:middle of one smaller, n minus 1. 00:41:54.206 --> 00:41:57.226 A:middle So, the formula for the nth term of the sequence, 00:41:57.856 --> 00:42:03.576 A:middle a sub n is equal to 5 times 2 to the power of n minus 1. 00:42:08.376 --> 00:42:11.436 A:middle Now, let's find the seventh term in the sequence. 00:42:12.076 --> 00:42:16.116 A:middle We have our formula, we're looking for a sub 7. 00:42:16.556 --> 00:42:21.486 A:middle So, we substitute 7 in for n. So, we have 5 times 2 00:42:21.486 --> 00:42:23.786 A:middle to the power of 7 minus 1. 00:42:24.266 --> 00:42:27.036 A:middle And what do we do first in our order of operations? 00:42:27.496 --> 00:42:29.106 A:middle We do the exponent first. 00:42:29.256 --> 00:42:31.356 A:middle So, 7 minus 1 is 6. 00:42:32.266 --> 00:42:33.776 A:middle Now, we take our calculator. 00:42:34.696 --> 00:42:35.986 A:middle Here's a calculator. 00:42:37.446 --> 00:42:39.896 A:middle Now, there's different ways you can do this. 00:42:40.346 --> 00:42:43.966 A:middle You could take 5 times -- 00:42:44.746 --> 00:42:47.496 A:middle your calculator knows the order of operations. 00:42:47.496 --> 00:42:50.106 A:middle Your calculator knows to do the exponents first. 00:42:50.136 --> 00:42:52.336 A:middle So, you could put in the base of 2. 00:42:53.166 --> 00:42:55.906 A:middle And this is the exponent key, 2 to the power 00:42:55.906 --> 00:42:59.326 A:middle of -- 2 to the power of 6. 00:43:00.186 --> 00:43:03.306 A:middle So, it knows to do exponents before multiplication, 00:43:03.466 --> 00:43:04.906 A:middle and then press Enter. 00:43:06.306 --> 00:43:08.356 A:middle We have an answer of 320. 00:43:08.946 --> 00:43:11.186 A:middle If you're not comfortable with this, 00:43:11.276 --> 00:43:13.286 A:middle let me show you another way to do the problem. 00:43:13.886 --> 00:43:15.716 A:middle Do the exponents first. 00:43:15.716 --> 00:43:17.006 A:middle You know the order of operations, 00:43:17.006 --> 00:43:18.546 A:middle you need to do the experiments first. 00:43:18.546 --> 00:43:22.446 A:middle So, you do the base 2 to the power of -- 00:43:22.676 --> 00:43:26.306 A:middle now, put in your exponent of 6, okay? 00:43:26.306 --> 00:43:29.466 A:middle We press Enter, and we have an answer of 64. 00:43:29.956 --> 00:43:32.276 A:middle So, 2 to the 6 is 64. 00:43:32.876 --> 00:43:34.886 A:middle Now, we're supposed to multiply that by 5. 00:43:34.886 --> 00:43:37.546 A:middle So, times 5. 00:43:38.666 --> 00:43:41.336 A:middle The calculator -- remember that I had 2 00:43:41.336 --> 00:43:42.826 A:middle to the 6 in the calculator. 00:43:43.646 --> 00:43:45.946 A:middle Now, we press Enter and we have 320. 00:43:47.006 --> 00:43:50.926 A:middle So, we have 2 to the 6 is 64 and then we multiply that by 5 00:43:51.106 --> 00:43:53.516 A:middle and we still get the answer of 320. 00:43:54.056 --> 00:43:56.736 A:middle The 7th term of the sequence is 320. 00:44:04.116 --> 00:44:06.366 A:middle Let's use the example we just finished to come 00:44:06.366 --> 00:44:10.586 A:middle up with a general formula for terms of a geometric sequence. 00:44:11.866 --> 00:44:17.316 A:middle We had a sub n equals 5 times 2 to the power of n minus 1. 00:44:17.776 --> 00:44:22.416 A:middle And in our sequence, a sub 1 was 5, and r was 2. 00:44:24.276 --> 00:44:29.456 A:middle Well, since a sub one is 5, we can replace 5 with a sub 1. 00:44:30.556 --> 00:44:34.426 A:middle And we can replace 2 with r. So, 00:44:34.426 --> 00:44:40.186 A:middle in a geometric sequence terms can be found using the formula a 00:44:40.186 --> 00:44:44.976 A:middle sub n equals a sub 1 times r to the n minus 1 power. 00:44:52.046 --> 00:44:54.826 A:middle Find a formula for the nth term of the following sequence. 00:44:55.226 --> 00:44:57.366 A:middle Use the formula to find the 9th term. 00:44:57.776 --> 00:45:03.026 A:middle Our sequence is 80, 40, 20, 10 and so on and so forth. 00:45:03.806 --> 00:45:06.566 A:middle So, we take 40 and divide it by the term 00:45:06.566 --> 00:45:08.536 A:middle that comes before it, which is 80. 00:45:09.346 --> 00:45:11.766 A:middle We reduce the fraction 40 over 80 00:45:11.806 --> 00:45:15.536 A:middle to the lowest terms and we get one half. 00:45:15.736 --> 00:45:19.016 A:middle Now, we take the term 20 and divide it by the term 00:45:19.016 --> 00:45:21.436 A:middle that comes before it, which is 40. 00:45:22.196 --> 00:45:26.626 A:middle 20 divided by 40 reduces also to one half. 00:45:27.606 --> 00:45:29.696 A:middle Since they both reduced to one half, 00:45:30.526 --> 00:45:32.666 A:middle this is a geometric sequence 00:45:32.996 --> 00:45:36.576 A:middle and the common ratio or r is one half. 00:45:37.096 --> 00:45:40.766 A:middle [variable] a sub 1 for our geometric sequence is 80, 00:45:40.926 --> 00:45:41.656 A:middle the first term. 00:45:42.866 --> 00:45:45.026 A:middle [variable] r, the common ratio is one half. 00:45:46.136 --> 00:45:48.126 A:middle Here's the formula we just came up with. 00:45:48.386 --> 00:45:52.606 A:middle [variable] a sub n equals a sub one the first term times r, 00:45:52.606 --> 00:45:55.666 A:middle the common ratio to the power of n minus 1. 00:45:56.466 --> 00:46:01.826 A:middle So, for our sequence, a sub 1 is 80, and r is one half. 00:46:02.056 --> 00:46:07.336 A:middle The formula for the nth term is a sub n equals 80 times the 00:46:07.336 --> 00:46:10.396 A:middle quantity one half to the n minus 1 power. 00:46:11.066 --> 00:46:14.886 A:middle In this case, you do need the one half to be in parenthesis, 00:46:15.356 --> 00:46:18.316 A:middle because you need the whole fraction to be raised 00:46:18.516 --> 00:46:19.976 A:middle to the n minus 1 power. 00:46:27.046 --> 00:46:29.486 A:middle Now, we wish to find the 9th term of the sequence. 00:46:30.206 --> 00:46:31.156 A:middle Here's our formula. 00:46:32.936 --> 00:46:34.006 A:middle We want a sub 9. 00:46:34.006 --> 00:46:37.106 A:middle So, we put in 9 for n, what do we do first? 00:46:37.586 --> 00:46:38.806 A:middle We do the exponent. 00:46:38.806 --> 00:46:40.176 A:middle So, 9 minus 1 is 8. 00:46:40.886 --> 00:46:42.386 A:middle Now, think order of operations, 00:46:42.716 --> 00:46:44.346 A:middle you need to do the exponents first. 00:46:44.346 --> 00:46:47.806 A:middle So, you need to take the 8th power of one half. 00:46:47.926 --> 00:46:50.296 A:middle Let's use a calculator to help us out with this. 00:46:50.416 --> 00:46:53.426 A:middle So, we want to take the 8th power of one half. 00:46:54.716 --> 00:46:57.236 A:middle So, on this calculator, it's going to look like this. 00:46:57.966 --> 00:47:01.906 A:middle Parenthesis, 1, and then we're going 00:47:01.906 --> 00:47:03.226 A:middle to use the fraction button. 00:47:04.446 --> 00:47:07.306 A:middle The fraction button on this calculator looks like this. 00:47:08.326 --> 00:47:13.186 A:middle On the TI-30XIIS, the buttons look like this. 00:47:13.186 --> 00:47:17.446 A:middle So, I'm going to show out the buttons on the TI-30XIIS also, 00:47:17.446 --> 00:47:18.366 A:middle because that's the one that a lot 00:47:18.366 --> 00:47:20.106 A:middle of students will use in our classes. 00:47:21.506 --> 00:47:25.316 A:middle So, parenthesis, 1 and then the fraction button. 00:47:27.006 --> 00:47:30.046 A:middle So, in this calculator, it actually looks like a fraction. 00:47:30.446 --> 00:47:33.446 A:middle And then you put in the 2 for the denominator. 00:47:34.816 --> 00:47:37.576 A:middle Now, we need to get out of the denominator. 00:47:38.326 --> 00:47:40.676 A:middle On our calculator, we can just close the parenthesis. 00:47:40.726 --> 00:47:42.926 A:middle But on this calculator, we have to get out of the parenthesis, 00:47:42.926 --> 00:47:44.456 A:middle so we have to use the right arrow key. 00:47:45.866 --> 00:47:49.956 A:middle So, now I close my parenthesis, okay? 00:47:49.956 --> 00:47:54.116 A:middle So, we have parenthesis, one half, close parenthesis. 00:47:55.096 --> 00:47:57.416 A:middle Now, we need to raise that to the 8th power. 00:47:57.996 --> 00:48:00.206 A:middle So, here's our power key. 00:48:00.206 --> 00:48:02.286 A:middle So, 2 to the power of 8. 00:48:02.286 --> 00:48:13.056 A:middle And then you can press Enter, and we get 1 over 256. 00:48:14.406 --> 00:48:19.576 A:middle So, one half raised to the 8th power is 1 over 256. 00:48:20.176 --> 00:48:22.426 A:middle We still need to multiply that times 80. 00:48:22.536 --> 00:48:25.546 A:middle The calculator remembers our previous result, right there. 00:48:25.546 --> 00:48:26.766 A:middle So, we don't have to reenter it. 00:48:27.216 --> 00:48:29.296 A:middle So, now I just press multiplication. 00:48:30.186 --> 00:48:34.996 A:middle And then we put in 80 and press Enter. 00:48:36.086 --> 00:48:39.256 A:middle And we get 80 over 256. 00:48:39.256 --> 00:48:42.046 A:middle But the calculator has already reduced that fraction. 00:48:42.946 --> 00:48:49.216 A:middle It took 80 over 256 and reduced it to 5 over 16. 00:48:50.046 --> 00:48:53.886 A:middle The 9th term of this sequence is 5/16. 00:49:01.046 --> 00:49:04.066 A:middle Find a formula for the nth term of the following sequence. 00:49:04.626 --> 00:49:06.846 A:middle Use the formula to find the 8th term. 00:49:07.236 --> 00:49:14.256 A:middle And our sequences 6, negative 18, 54, negative 162, 00:49:14.376 --> 00:49:15.456 A:middle and so on and so forth. 00:49:16.046 --> 00:49:19.426 A:middle So, we take negative 18 and divide it 00:49:19.456 --> 00:49:21.216 A:middle by the first term, which is 6. 00:49:21.886 --> 00:49:23.536 A:middle That reduces to negative 3. 00:49:24.766 --> 00:49:28.456 A:middle We take 54 and divide it by negative 18. 00:49:29.106 --> 00:49:31.876 A:middle And we get negative 3 also. 00:49:32.666 --> 00:49:34.906 A:middle So, this is a geometric sequence. 00:49:36.026 --> 00:49:38.456 A:middle And the common ratio is negative 3. 00:49:39.426 --> 00:49:41.136 A:middle So, a sub 1 is 6. 00:49:41.136 --> 00:49:42.446 A:middle The first term is 6. 00:49:42.966 --> 00:49:46.666 A:middle This is geometric, and the common ratio is negative 3. 00:49:47.556 --> 00:49:51.726 A:middle The formula for a geometric sequence, the nth term is this, 00:49:52.126 --> 00:49:55.066 A:middle a sub n equals a sub 1 times r to the n minus one. 00:49:56.366 --> 00:49:58.066 A:middle The formula for the nth term 00:49:58.066 --> 00:50:03.036 A:middle of this sequence is a sub equals 6 times the quantity negative 3 00:50:03.036 --> 00:50:03.976 A:middle to the n minus 1. 00:50:04.616 --> 00:50:06.976 A:middle The negative 3 definitely needs to be in parenthesis. 00:50:13.246 --> 00:50:16.846 A:middle Now, we wish to find the 8th term of this sequence. 00:50:17.536 --> 00:50:19.076 A:middle We have our formula. 00:50:20.236 --> 00:50:25.296 A:middle We substitute in 8 for n. We look at the exponent first. 00:50:25.546 --> 00:50:26.996 A:middle 8 minus 1 is 7. 00:50:28.076 --> 00:50:30.686 A:middle And now, we're going to need our calculator to help us out. 00:50:31.816 --> 00:50:33.366 A:middle You can do this different ways. 00:50:34.396 --> 00:50:40.076 A:middle You could do 6 times parenthesis -- 00:50:40.076 --> 00:50:45.426 A:middle now, do I hit this key for subtraction? 00:50:46.006 --> 00:50:49.966 A:middle Kind of giving it away, or this key for negative? 00:50:50.516 --> 00:50:51.346 A:middle Negative, good. 00:50:51.706 --> 00:50:54.536 A:middle Negative, what is it? 00:50:54.766 --> 00:50:58.096 A:middle Three. Close parenthesis. 00:50:59.036 --> 00:51:02.996 A:middle And then to the power of 7, to the power of 7. 00:51:03.466 --> 00:51:06.146 A:middle So, in this way, I put the whole thing 00:51:06.146 --> 00:51:07.756 A:middle in straight from left to right. 00:51:08.116 --> 00:51:10.926 A:middle 6 times negative 3 to the 7th. 00:51:10.926 --> 00:51:14.836 A:middle The calculator knows to do the exponent first. 00:51:14.836 --> 00:51:17.856 A:middle It's going to take the 7th power of negative three first, 00:51:18.526 --> 00:51:21.326 A:middle and then it's going to multiply it by 6. 00:51:22.066 --> 00:51:26.106 A:middle So, there's our answer, negative 13,122. 00:51:27.296 --> 00:51:30.666 A:middle Here's another way to do it, parenthesis, 00:51:31.216 --> 00:51:34.366 A:middle negative 3, close parenthesis. 00:51:34.696 --> 00:51:38.736 A:middle We're going to raise that to the 7th power, okay? 00:51:38.876 --> 00:51:44.346 A:middle So, we raise that to the 7th power and we get negative 2,187. 00:51:44.776 --> 00:51:48.106 A:middle And then we're going to take our answer and multiply it times 6. 00:51:48.486 --> 00:51:51.976 A:middle So, times 6 equals. 00:51:53.436 --> 00:51:55.406 A:middle Do it the way that you're most comfortable with. 00:51:55.916 --> 00:52:00.976 A:middle Either way, the 8th term of this sequence is negative 13,122. 00:52:07.046 --> 00:52:07.976 A:middle Pause the video and try these problems. 00:52:08.516 --> 00:52:19.500 A:middle [ Pause ]