Consider a real valued function $f(x)$ satisfying, $2f(xy) = (f(x))^y + (f(y))^x$ for all real $x\ \mbox{\&}\ y$ and $f(1) = a$ where $a\neq 1$. Prove that $(a - 1)\sum_{i = 1}^{n}f(i) = a^{n + 1} - a$.
Consider a real valued function $f(x)$ satisfying, $2f(xy) = (f(x))^y + (f(y))^x$ for all real $x\ \mbox{\&}\ y$ and $f(1) = a$ where $a\neq 1$. Prove that $(a - 1)\sum_{i = 1}^{n}f(i) = a^{n + 1} - a$.
Taking y= 1, $2f(x)= f(x)+ f(1)^x= f(x)+ a^x$. That is, $f(x)= a^x$. That should be all you need.