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  4. The sum to n terms of the series (1/2)+(3/4)+(7/8)+(15/16)+ ldots ldots. is equal to

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

Sequences and Series

Solution:

$S=\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots . . n$ terms
$=\left(1-\frac{1}{2}\right)+\left(1-\frac{1}{2^2}\right)+\left(1-\frac{1}{2^3}\right)+\ldots \ldots \ldots . .+\left(1-\frac{1}{2^n}\right)=n-\left(\frac{1}{2}+\frac{1}{2^2}+\ldots \ldots \cdot \frac{1}{2^n}\right)$
$=n-\frac{1}{2} \frac{1-\left(\frac{1}{2}\right)^n}{1-\left(\frac{1}{2}\right)}=n+2^{-n}-1$