A recursive sequence is a sequence that instead of having a set formula where you're multiplying by a current rate to get the next terms or adding something to get the next terms, what you’re doing is actually using the previous one or two or even three terms to somehow get the next term.
So what we’re going to do is we have some information regarding a sequence and we’re going to just write out the first couple of terms. What we have is we know that the first term is 1, second term is 2 and we know that a sub n is equal to 2, a sub n minus 1 plus a sub n minus 2.
When dealing with recursive formulas I often times have a really hard time seeing what all this stuff, means the a sub 1, a sub n plus 1, a sub n minus 2, so whenever I’m trying to find a term I just actually write in the n's that I want and then things become a little a bit more straight forward.
The next three terms, that means I want to find a sub 3, 4 and 5. So a sub 3 then. We’re plugging 3 in for n, and what I’m going to do is plug in 3 for all the n's and then go back and see what that means. This is equal to 2 a sub 3 minus 1. This is just going to be sub 2, plus plugging in 3 for this end as well, a sub 1.
Now this becomes a little more clear to me as to what’s going on. All I’m taking is I’m plugging these terms in, a sub 2 is just 2. So 2 times 2 is 4, plus a sub 1 which is 1 so we end up with 5. So I found a sub 3. Going though the same process for a sub 4.
We’re plugging 4 in and we are subtracting 1, so this term becomes a sub 3 plus plug 4 and subtract 2, this just becomes a sub 2. We know a sub 3 is 5, so 2 times 5 is 10. We know a sub 2 is 2 so 10 plus 2 just becomes 12. Repeating for a sub 5, 2 times 5 minus 1, so this is a sub 4 plus 5 minus 2, a sub 3, and what we end up with is 2 times a sub 4 which is 12, so 2 times 12 is 24 plus a sub 3 which is 5, 24 plus 5 is 29.
Using a recursion formula to write out terms of a sequence. It looks pretty complicated when you’re dealing with a lot of a sub n minus things, but if you write it out, just replace your n's with the term number you’re looking for, life become pretty straight forward and then you just plug in to evaluate.