<s>[INST] Good idea to find below sum this is the question.
Purpose is to find sum of $$f\left(\frac{1}{14}\right)+f\left(\frac{2}{14}\right)+f\left(\frac{3}{14}\right)+...+f\left(\frac{13}{14}\right)$$for function $$f(x)=\frac{4^x}{4^x+2}$$Is there a nice method to find $f\left(\frac{1}{14}\right)+f\left(\frac{2}{14}\right)+f\left(\frac{3}{14}\right)+...+f\left(\frac{13}{14}\right)$ ?
 [/INST] Yup, a nice thing to notice is the value of
$$
\begin{align}
f(x)+f(1-x)
&=\frac{4^x}{4^x+2}+\frac{4}{4+2\cdot4^x}\\
&=\frac{4^x}{4^x+2}+\frac{2}{4^x+2}\\[3pt]
&=\cdots\,?
\end{align}
$$
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