SATHEE: Sequences and Series 4 Question 9

Sequences and Series 4 Question 9

9. Sum of the first $n$ terms of the series $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \dots$ 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 $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \dots$ upto $n$ terms can be written as

$\begin{aligned} 1 - \frac{1}{2} + 1 - \frac{1}{4} + 1 - \frac{1}{8} + 1 - \frac{1}{16} + \dots upto n terms \\ = n - \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + n terms \\ = n - \frac{\frac{1}{2} 1 - \frac{1}{2^{n}}}{1 - \frac{1}{2}} = n + 2^{- n} - 1 \end{aligned}$

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Sequences and Series 4 Question 9

9. Sum of the first n terms of the series 12+34+78+1516+… is equal to

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9. Sum of the first $n$ terms of the series $\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \dots$ is equal to