First slide

Special Series in Sequences and Series

Question

$The sum 1 + 3 + 7 + 15 + 31 + \dots to 100 terms is$

Moderate

A

$2^{100} - 102$
B

$2^{99} - 101$
C

$2^{102} - 103$
D

$2^{101} - 102$

Solution

First difference are in G.P. $S = (2^{1} - 1) + (2^{2} - 1) + (2^{3} - 1) + \dots . + (2^{100} - 1) \frac{2 (2^{100} - 1)}{2 - 1} - 100 = 2^{101} - 102$

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