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## AP®︎ Calculus BC (2017 edition)

### Course: AP®︎ Calculus BC (2017 edition)>Unit 12

Lesson 1: Infinite sequences

# Worked example: sequence recursive formula

Using the recursive formula of a sequence to find its fifth term. Created by Sal Khan.

## Video transcript

- [Instructor] A sequence is defined recursively as follows. So A sub N is equal to A sub N minus one times A sub N minus two or another way of thinking about it. the Nth term is equal to the N minus oneth term times the N minus two-th term with the zeroth term where A sub zero is equal to two and A sub one is equal to three. Find A sub four. So let's write this down. They're telling us A sub zero is equal to two and they also tell us that A sub one is equal to three. So they've kind of given us our starting conditions or our base conditions. Now we can think about what A sub two is. They tell us that A sub two is going to be A sub two minus one, so that's A sub one. It's A sub one times A sub two minus two. That's A sub zero. A times A sub zero. They already told us what A sub one and A sub zero is. This thing is three, this thing is two. It's three times two which is equal to six. Now let's move on to A sub three. A sub three is going to be the product of the previous two terms. It's going to be A sub two, three minus one is two, three minus two is one. So it's A sub two. A sub two times A sub one. Times A sub one. So it's equal to six times three six times three. which is equal to 18. Which is equal to 18. And then finally A sub four, which I will do in a color that I'll use, I'll do it in yellow. A sub four is going to be equal to A sub three, A sub three times A sub two. So four minus one is three, four minus two is two. So times A sub two, times A, and then a blue color. Times A sub two, which is equal to 18, 18 times six. 18 times six which is equal to, let's see, six times eight is 48 plus 60, or six times 10 is 100, 108. And we're done, A sub four is equal to 108.