First slide

Arithmetico-Geometric Progression

Question

The positive integer n for which $2 \times 2^{2} + 3 \times 2^{3} + 4 \times 2^{4} + \dots + n \times 2^{n} = 2^{n + 10}$ is

Moderate

A

510
B

511
C

512
D

513

Solution

We have
$\begin{array}{l} 2^{n + 10} = 2 \times 2^{2} + 3 \times 2^{3} + 4 \times 2^{4} + \dots + n \times 2^{n} \\ \Rightarrow 2 (2^{n + 10}) = 2 \times 2^{3} + 3 \times 2^{4} + \dots + (n - 1) \times 2^{n} + n \times 2^{n + 1} \end{array}$
Subtracting, we get
$\begin{array}{l} - 2^{n + 10} = 2 \cdot 2^{2} + 2^{3} + 2^{4} + \dots + 2^{n} - n \cdot 2^{n + 1} \\ = 8 + \frac{8 (2^{n - 2} - 1)}{2 - 1} - n \times 2^{n + 1} \\ = 2^{n + 1} - (n) 2^{n + 1} \\ \Rightarrow 2^{10} = 2 n - 2 \Rightarrow n = 513 \end{array}$

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