First slide

Special Series in Sequences and Series

Question

The sum of the series $1 + 2 \cdot 2 + 3 \cdot 2^{2} + 4 \cdot 2^{3} + 5 \cdot 2^{4} + \dots + 100 \cdot 2^{99} is$

Moderate

A

$99 \cdot 2^{100} - 1$
B

$100 \cdot 2^{100}$
C

$99 \cdot 2^{100}$
D

$99 \cdot 2^{100} + 1$

Solution

Let $S = 1 + 2 \cdot 2 + 3 \cdot 2^{2} + 4 \cdot 2^{3} + 5 \cdot 2^{4} + \dots + 100 \cdot 2^{99}$
$∴ 2 S = 1 \cdot 2 + 2 \cdot 2^{2} + 3 \cdot 2^{3} + \dots + 99 \cdot 2^{99} + 100 \cdot 2^{100}$
Substracting, we get
$\begin{array}{l} - S = 1 + 1 \cdot 2 + 1 \cdot 2^{2} + \dots + 1 \cdot 2^{99} - 100 \cdot 2^{100} \\ = (1 + 2 + 2^{2} + \dots + 2^{99}) - 100 \cdot 2^{100} \\ = \frac{1 (2^{100} - 1)}{2 - 1} - 100 \cdot 2^{100} = 2^{100} - 1 - 100 \cdot 2^{100} \\ ∴ S = 100 \cdot 2^{100} - 2^{100} + 1 = 99 \cdot 2^{100} + 1 . \end{array}$

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