MCQ
The value of $\sum \limits_{n=0}^{1947} \frac{1}{2^n+\sqrt{2^{1994}}}$ is equal to
- ✓$\frac{487}{\sqrt{2^{1945}}}$
- B$\frac{1946}{\sqrt{2^{1947}}}$
- C$\frac{1947}{\sqrt{2^{1947}}}$
- D$\frac{1948}{\sqrt{2^{1947}}}$
We have,
$\sum \limits_{n=0}^{1947} \frac{1}{2^n+\sqrt{2^{1947}}}$
Let $\quad f(n)=\frac{1}{2^n+\sqrt{2^{1947}}}$
$f(0)+f(1947)=\frac{1}{1+2^{\frac{1947}{2}}}+\frac{1}{2^{1947}+2^{\frac{1947}{2}}}$
$=\frac{1}{1+2^{\frac{1947}{2}}}+\frac{1}{2^{\frac{1947}{2}}\left(2^{\frac{1947}{2}}+1\right)}$
$=\frac{2^{\frac{1947}{2}}+1}{2^{\frac{1947}{2}}\left(2^{\frac{1947}{2}+1}\right)}=\frac{1}{2^{\frac{1947}{2}}}$
Similarly, $f(1)+f(1946)=\frac{1}{2^{\frac{1947}{2}}}$
$\because \sum \limits_{n=0}^{1947} f(x)=f(0)+f(1)+f(2)+f(3)+\ldots .+f(1947)$
$=(f(0)+f(1947)+(f(1)+f(1946))+\ldots +(f(973)+f(974))$
$=974 \times \frac{1}{2^{\frac{1917}{2}}}$
$\sum \limits_{n=0}^{1947} f(n)=\frac{2 \times 487}{2 \times 2^{\frac{1945}{2}}}=\frac{487}{\sqrt{2^{1945}}}$
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