Sequences and Series 4 Question 9
9. Sum of the first $n$ terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots$ is equal to
(1988, 2M)
(a) $2^{n}-n-1$
(b) $1-2^{-n}$
(c) $n+2^{-n}-1$
(d) $2^{n}+1$
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Solution:
- Sum of the $n$ terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots$ upto $n$ terms can be written as
$$ \begin{aligned} & 1-\frac{1}{2}+1-\frac{1}{4}+1-\frac{1}{8}+1-\frac{1}{16}+\ldots \text { upto } n \text { terms } \\ &=n-\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots+n \text { terms } \\ &=n-\frac{\frac{1}{2} 1-\frac{1}{2^{n}}}{1-\frac{1}{2}}=n+2^{-n}-1 \end{aligned} $$
