Prove the sequence converge and find its limit

Consider the following sequence:

$a_{n+2} = \frac{a_{n+1}+a_{n}}{2}$, $a_{1}$ and $a_{2}$ are given.

Write $a_{n}$ as a function of $a_{1}$ and $a_{2}$ and show that its limit is $\frac{a1 + 2a_{2}}{3}$

I think I am loosing myself on algebra here. I can't even do the first part. Any help is welcome, thanks a lot!

Solution 1:

$A = \pmatrix{1/2&1/2\\1&0}$

$\pmatrix{a_{n}\\a_{n-1}}=A\pmatrix{a_{n-1}\\a_{n-2}}$

$\pmatrix{a_{n+2}\\a_{n+1}}=A^n\pmatrix{a_2\\a_1}$

$A = PDP^{-1}; A^n =PD^nP^{-1}$

$A = \pmatrix{1&1\\1&-2}\pmatrix{1&0\\0&-1/2}\pmatrix{2/3&1/3\\1/3&-1/3}$

$A^n = \pmatrix{1&1\\1&-2}\pmatrix{1&0\\0&-2^{-n}}\pmatrix{2/3&1/3\\1/3&-1/3}$

Limit as $n\to \infty = \pmatrix{1&1\\1&-2}\pmatrix{1&0\\0&0}\pmatrix{2/3&1/3\\1/3&-1/3}$

$\pmatrix{2/3&1/3\\2/3&1/3}$

$\lim\limits_{n\to\infty} a_n = 2/3a_2 + 1/3 a_1$