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BITSAT 2016
Exam
BITSAT
Year
2016
Subject
Mathematics
Chapter
Sequences and Series

BITSAT 2016 · Mathematics · Sequences and Series

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

The sum to n\mathrm{n} terms of the series 12+34+78+1516+…………\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots \ldots \ldots \ldots is

Options

  1. An−1−2−nn-1-2^{-n}
  2. B1
  3. Cn−1+2−nn-1+2^{-n}
  4. D1+2−n1+2^{-n}

Correct answer

C. n−1+2−nn-1+2^{-n}

Step-by-step solution

12+34+78+1516+…=(1−12)+(1−14)+(1−18)+(1−116)+….=n−12{1−12n}1−12=n−1+2−n\begin{aligned} \frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots \\=&\left(1-\frac{1}{2}\right)+\left(1-\frac{1}{4}\right)+\left(1-\frac{1}{8}\right)+\left(1-\frac{1}{16}\right)+\ldots. \\=& \mathrm{n}-\frac{\frac{1}{2}\left\{1-\frac{1}{2^{\mathrm{n}}}\right\}}{1-\frac{1}{2}}=\mathrm{n}-1+2^{-\mathrm{n}} \end{aligned}
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