Hi I am having a bit of trouble with a couple of questions in regards to convergences.

Let the sequence $\displaystyle \{a_{n}\}^\infty _{1}$

let $\displaystyle a_{1} =1, a_{2} = 2, a_{n+2} = \frac{a_{n+1} + a_{n}}{2}$

Prove that $\displaystyle |a_{n+2} - a_{n+1}| = \frac {1}{2^n}$

Hence prove that $\displaystyle \{a_{n}\}^\infty_{1}$

From my attempts at part 1 I have been able to show that for n=0 and n=1 that the function will be equal to $\displaystyle \frac{1}{2^0}$ and $\displaystyle \frac{1}{2^1}$ respectively. However I am having trouble actually proving for n terms that it is true. I am able to get to $\displaystyle |\frac{a_{n+1} - a_{n-1}}{2}|$ but am completely stuck there on.

For the second part I have very little idea as to how I can prove that the sequence converges using my result from part 1.

Thanks you for any input in advance :d