Worked example: sequence recursive formula | Series | AP Calculus BC | Khan Academy

A sequence is defined recursively as follows: so a sub n is equal to a sub n minus 1 times a sub n minus 2.

Or another way of thinking about it, the nth term is equal to the n minus 1 term times the n minus 2th term. With this, the zeroth term, or a sub zero, is equal to 2, and a sub one is equal to 3.

Find a sub four. So let's write this down. They're telling us a sub zero is equal to 2, and they also tell us that a sub 1 is equal to 3.

So they've kind of given us our starting conditions or our base conditions. Now we can think about what a sub 2 is, and they tell us that a sub 2 is going to be a sub 2 minus 1, so that's a sub 1. It's a sub 1 times a sub 2 minus 2; that's a sub 0.

So a times a sub 0. They already told us what a sub 1 and a sub 0 is. This thing is 3; this thing is 2, so it's 3 times 2, which is equal to 6.

Now let's move on to a sub 3. So a sub 3 is going to be the product of the previous two terms. So it's going to be a sub 2; 3 minus 1 is 2, and 3 minus 2 is 1.

So it's a sub 2 times a sub 1, which is equal to 6 times 3. 6 times 3, which is equal to 18.

Now finally, a sub 4, which I will do in a color that I'll use; I'll do it in yellow. A sub 4 is going to be equal to a sub 3; a sub 3 times a sub 2.

So notice that 4 minus 1 is 3 and 4 minus 2 is 2. So it’s times a sub 2, which is equal to 18 times 6.

18 times 6, which is equal to... let's see, 6 times 8 is 48 plus 60 is... or 6 times 10 is one hundred, one hundred and eight, and we're done.

A sub four is equal to one hundred and eight.