Sequence And Series Question 608
Question: The sum of the first $ n $ terms of the series $ \frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+……… $ is
[IIT 1988; MP PET 1996; RPET 1996, 2000; Pb. CET 1994; DCE 1995, 96]
Options:
A) $ 2^{n}-n-1 $
B) $ 1-{2^{-n}} $
C) $ n+{2^{-n}}-1 $
D) $ 2^{n}-1 $
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Answer:
Correct Answer: C
Solution:
The sum of the first $ n $ terms is $ S_{n}=( 1-\frac{1}{2} )+( 1-\frac{1}{2^{2}} )+( 1-\frac{1}{2^{3}} )+( 1-\frac{1}{2^{4}} ) $ $ +……+( 1-\frac{1}{2^{n}} ) $ $ =n-{ \frac{1}{2}+\frac{1}{2^{2}}+…..+\frac{1}{2^{n}} } $ = $ n-\frac{1}{2}( \frac{1-\frac{1}{2^{n}}}{1-\frac{1}{2}} )=n-( 1-\frac{1}{2^{n}} )=n-1+{2^{-n}} $ . Trick: Check for $ n=1,\ 2\ i.e. $ $ S_1=\frac{1}{2},\ S_2=\frac{5}{4} $ and (c)
$ \Rightarrow S_1=\frac{1}{2} $ and $ S_2=2+{2^{-2}}-1=\frac{5}{4} $ .
BETA
