Sumo and Easy Sum

\[S=\displaystyle\sum_{i=1}^{+\infty}\frac{fib[i]}{k^i} \]

\[kS=\displaystyle\sum_{i=1}^{+\infty}\frac{fib[i+1]}{k^i}+fib[1] \]

两式相加

\[(k+1)S=\displaystyle\sum_{i=1}^{+\infty}\frac{fib[i]}{k^i}+\displaystyle\sum_{i=1}^{+\infty}\frac{fib[i+1]}{k^i}+fib[1] \]

\[(k+1)S=\displaystyle\sum_{i=1}^{+\infty}\frac{fib[i+2]}{k^i}+fib[1] \]

\[\frac{(k+1)S}{k^2}=\displaystyle\sum_{i=1}^{+\infty}\frac{fib[i+2]}{k^{i+2}}+\frac{fib[1]}{k^2} \]

\[\frac{(k+1)S}{k^2}=\displaystyle\sum_{i=1}^{+\infty}\frac{fib[i]}{k^{i}}-\frac{fib[1]}{k}-\frac{fib[2]}{k^2}+\frac{fib[1]}{k^2} \]

\[\frac{(k+1)S}{k^2}=S-\frac{fib[1]}{k}-\frac{fib[2]}{k^2}+\frac{fib[1]}{k^2} \]

\[(k+1)S=k^2S-k*fib[1]-fib[2]+fib[1] \]

\[S=\frac{k}{k^2-k-1} \]