pithon_with_an_i wrote: For all $n \geq 1$, define $a_{n}$ to be the fraction $\frac{k}{2^n}$ such that $a_{n}$ is the closest to $\frac{1}{3}$ over all integer values of $k$. Prove that the sequence $a_{1}, a_{2}, \cdots $satisfies the equation $2a_{i+2} = a_{i+1} + a_{i}$ for all $i \geq 1$. Easy to get $a_n=\frac{2^n-(-1)^n}{3.2^n}$ and from there to check that indeed $2a_{i+2} = a_{i+1} + a_{i}$

Exact same as above.