i need help rewriting this recursive sequence

[rugged glacier]

i need help rewriting this recursive sequence into a formular that is not recursive (sry for my bad english)

[viral halo]

You mean you need the algorithm to get from the first expression to the second?

[rugged glacier]

up above i wrote the sequence and the recursive definition, i also know that the formular im searching for looks like the one on the bottom, i just dont know how to get there on my own

[rugged glacier]

yes

[viral halo]

I assume the initial conditions are a(1) = 1, a(2) = 1.
a(n) = a(n - 1) + 2a(n - 2), a(1) = 1
First, let's collect all the terms on one side.
a(n) - a(n - 1) - 2a(n - 2) = 0
Then let's write the characteristic equation.
λ^2 - λ - 2 = 0
(λ - 2)(λ + 1) = 0
The roots are λ = -1, λ = 2.
So, the general solution is (A and B are constants):
a(n) = A 2^n + B(-1)^n
Now, let's apply the initial conditions.
a(1) = 2A - B = 1
a(2) = 4A + B = 1
Solving this system gives A = 1/3, B = -1/3. So, the particular solution is:
a(n) = (1/3)2^n - (1/3)(-1)^n = (2^n - (-1)^n)/3

[rugged glacier]

oh wow

i never thought of applying the characteristic equation

[rugged glacier]

thanks mate

[candid coyote]

@rugged glacier has given 1 rep to @viral halo

[viral halo]

You're welcome!

[fresh lion]

✅ This help channel has been closed

[rugged glacier]

is there something you cant solve xD

[viral halo]

Well, of course. I am a chemistry major, so I haven't learned stuff like number theory, complex analysis (unfortunately, as that could actually be helpful), etc.

[rugged glacier]

im very impressed by your math skills!

[viral halo]

Ah, thanks 😄