Q58 / 170 · Manipal (MET) 2018 · Mathematics

The sum to $n$ terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots$ is

  • A
    $2^n-1$
  • B
    $1-2^n$
  • C
    $n+2^n-1$
  • Correct answer
    D
    $n-1+2^{-n}$

Solution

$\begin{aligned} & \frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots \text { upto } n \text { terms } \\ & =\left(1-\frac{1}{2}\right)+\left(1-\frac{1}{4}\right)+\left(1-\frac{1}{8}\right) \ldots \text { upto } n \text { terms } \\ & =(1+1+1+\ldots n \text { terms }) \\ & \quad-\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\ldots n \text { terms }\right) \\ & =n-\frac{\frac{1}{2}\left(1-\frac{1}{2^n}\right)}{1-\frac{1}{2}}=n-1+\frac{1}{2^n}=n-1+2^{-n}\end{aligned}$

Hence, the correct answer is option: 'D'.

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