If f(x)=2^x2^x+√2,x R, then _k=1^81f((k)/(82)) is equal to

Question

If $f(x)=\frac{{2}^{x}}{{2}^{x}+\sqrt{2}},x\in R$, then $\sum _{k=1}^{81}f(\frac{k}{82})$ is equal to

Accepted Answer

$$\begin{aligned}
& f(x)=\frac{2^x}{2^x+\sqrt{2}} \
& f(x)+f(1-x)=\frac{2^x}{2^x+\sqrt{2}}+\frac{2^{1-x}}{2^{1-x}+\sqrt{2}} \
& =\frac{2^x}{2^x+\sqrt{2}}+\frac{2}{2+\sqrt{2} 2^x}=\frac{2^x+\sqrt{2}}{2^x+\sqrt{2}}=1
\end{aligned}$$

$$\[\begin{gathered} \text { Now, } \sum_{\mathrm{k}=1}^{81} \mathrm{f}\left(\frac{\mathrm{k}}{82}\right)=\mathrm{f}\left(\frac{1}{82}\right)+\mathrm{f}\left(\frac{2}{82}\right)+\ldots \ldots+\mathrm{f}\left(\frac{81}{82}\right) \ =\mathrm{f}\left(\frac{1}{82}\right)+\mathrm{f}\left(\frac{2}{82}\right)+\ldots \ldots .+\mathrm{f}\left(1-\frac{2}{82}\right)+\mathrm{f}\left(1-\frac{1}{82}\right) \ =\left[\mathrm{f}\left(\frac{1}{82}\right)+\mathrm{f}\left(1-\frac{1}{82}\right)\right]+\left[\mathrm{f}\left(\frac{2}{82}\right)+\mathrm{f}\left(1-\frac{2}{82}\right)\right]+\ldots . .40 \text { cases }+\mathrm{f}\left(\frac{41}{82}\right) \end{gathered}$$\]

$$\begin{aligned}
& =(1+1+\ldots 40 \text { times })+\frac{2^{1 / 2}}{2^{1 / 2}+2^{1 / 2}} \
& 40+\frac{1}{2}=\frac{81}{2}
\end{aligned}$$

Solutions · 1