Introducing myself + recurrence problem

Hi. I'm Paul from Romania. I joined this forum because i wanted to share my knowledge and learn from you all.

I have a small problem.

Is there any particular formula to find the general term of a recurrence relation?

I must find it for this one:

a_{1}=1

a_{2}=5

a_{n+2}=a_{n+1} + 2a_{n}

--------------------

a_{n}=?

Re: Introducing myself + recurrence problem

Re: Introducing myself + recurrence problem

Hello, Paul!

Welcome aboard!

Here is a rather primitive method for this problem.

We have: .

We conjecture that: .

. . (The general term is exponential in nature.)

Then we have: .

Divide by

Hence: .

The function seems to be: .

Assume that is a *linear** *__combination__ of these two functions.

. . That is: .

We know the first two terms of the sequence:

. .

Add [1] and [2]: .

Substitute into [1]: .

Therefore: .