Supremum and infimum of $\sum_{k=1}^{n}\frac{a_k}{a_{k}+a_{k+1}+a_{k+2}}$

Question

Let, $n\ge 3$ be a fixed integer. Take all the possible finite sequences $(a_1,a_2,..,a_n)$ of positive numbers. Find, supremum and infimum of $$\sum_{k=1}^{n}\frac{a_k}{a_{k}+a_{k+1}+a_{k+2}}$$ where, $a_{n+1}=a_1,a_{n+2}=a_2$

My partial progress: Let, $s=a_1+a_2+..+a_n.$ Since, $$\frac{a_k}{s}\le\frac{a_k}{a_k+a_{k+1}+a_{k+2}}\le 1-\frac{a_k}{s}-\frac{a_{k+1}}{s}$$ which gives, $$1\le\sum_{k=1}^{n}\frac{a_k}{a_k+a_{k+1}+a_{k+2}}\le n-2$$ I guess, $n-2$ and $1$ will be the supremum and infinmum respectively,but I failed to show that.

Answer 1

Put $a_k=m^k$, when $m\rightarrow \infty $ the sum approaches $1$.

Put $a_k=m^{-k}$, when $m\rightarrow \infty $ the sum approaches $n-2$.