જો f(x)=frac{2^x}{2^x+sqrt{2}}, x in R, હોય, | vedclass

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Question

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

Vedclass · Accepted Answer

$\frac{81}{2}$

Vedclass · Answer

$f(x)=\frac{2^x}{2^x+\sqrt{2}}$
$f(x)+f(1-x)=\frac{2^x}{2^x+\sqrt{2}}+\frac{2^{1-x}}{2^{1-x}+\sqrt{2}}$
$=\frac{2^{x}}{2^{x}+\sqrt{2}}+\frac{2}{2+\sqrt{2} 2^{x}}=\frac{2^{x}+\sqrt{2}}{2^{x}+\sqrt{2}}=1$
Now, $\sum_{k=1}^{81} f\left(\frac{k}{82}ight)=f\left(\frac{1}{82}ight)+f\left(\frac{2}{82}ight)+\ldots \ldots .+f\left(\frac{81}{82}ight)$
$=f\left(\frac{1}{82}ight)+f\left(\frac{1}{82}ight)+\ldots \ldots+f\left(1-\frac{2}{82}ight)+f\left(1-\frac{1}{82}ight)$
${\left[f\left(\frac{1}{82}ight)+f\left(1-\frac{1}{82}ight)ight]+\left[f\left(\frac{2}{82}ight)+f\left(1-\frac{2}{82}ight)ight]+\ldots .40 	ext { cases }+f\left(\frac{41}{82}ight)}$
$(1+1+\ldots . .+1) 40 	ext { times }+\frac{2^{1 / 2}}{2^{1 / 2}+2^{1 / 2}}$
$40+\frac{1}{2}=\frac{81}{2}$

જો $f(x)=\frac{2^x}{2^x+\sqrt{2}}, x \in R$, હોય, તો $\sum_{k=1}^{81} f\left(\frac{k}{82}\right)$ $=$______.

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