<s>[INST] Sum up a function series $f(1/9)+f(2/9)+\dots+f(26/9)$ for $f(x)=\frac{9^{x}}{9^{x}+27}$ Given $f(x)=\dfrac{9^{x}}{9^{x}+27}$.
Find:
$$S=f\left(\frac{1}{9}\right)+f\left(\frac{2}{9}\right)+\dotsb+ f\left(\frac{26}{9}\right)$$
Teacher did not allow us to use calculator...Use sigma notation???
 [/INST] Observe that $f(x)=\frac{1}{1+\frac{27}{9^x}}$ and $f(3-x)=\frac{1}{1+\frac{9^x}{27}}$. Adding these two expressions gives $f(x)+f(3-x)=1$. (It may be easier to see this by making a substitute such as $a=\frac{27}{9^x}$ and $a^{-1}=\frac{9^x}{27}$). Therefore, $$S=f\left(\frac{1}{9}\right)+f\left(\frac{2}{9}\right)+\dotsb+ f\left(\frac{26}{9}\right)=13.$$
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