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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}=?
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: .