The positive integer n for which 2×22+3×23+4×24+⋯+n×2n=2n+10 is

2×22+3×23+4×24+⋯+n×2n=2n+10⇒1×21+2×22+3×23+⋯+n×2n=2n+10+2→(1)1×22+2×23+⋯+(n−1)2n+n.2n+1=2[2n+10+2]→(2)(1)−(2)21+22+23+24+⋯+2n−n.2n+1=2n+10+2−2n+11−421+22+23+24+⋯+2n=2n+10+2−2n+10.2−4+n.2n+1⇒2(2n−1)2−1=2n+10(1−2)−2+n.2n+1⇒2(2n−1)2−1=n.2n+1−2n+10−2∴2n+10−2=n.2n+1−2n+10−2⇒2n+10=2n+1(n−1)⇒29=n−1⇒n=513

The positive integer n for which 2×22+3×23+4×24+⋯+n×2n=2n+10 is

2×22+3×23+4×24+⋯+n×2n=2n+10⇒1×21+2×22+3×23+⋯+n×2n=2n+10+2→(1)1×22+2×23+⋯+(n−1)2n+n.2n+1=2[2n+10+2]→(2)(1)−(2)21+22+23+24+⋯+2n−n.2n+1=2n+10+2−2n+11−421+22+23+24+⋯+2n=2n+10+2−2n+10.2−4+n.2n+1⇒2(2n−1)2−1=2n+10(1−2)−2+n.2n+1⇒2(2n−1)2−1=n.2n+1−2n+10−2∴2n+10−2=n.2n+1−2n+10−2⇒2n+10=2n+1(n−1)⇒29=n−1⇒n=513

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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

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511

512

513

510

answer is D.

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Detailed Solution

$2 \times 2^{2} + 3 \times 2^{3} + 4 \times 2^{4} + \dots + n \times 2^{n} = 2^{n + 10}$

$\begin{array}{l} \Rightarrow 1 \times 2^{1} + 2 \times 2^{2} + 3 \times 2^{3} + \dots + n \times 2^{n} = 2^{n + 10} + 2 \to (1) \\ 1 \times 2^{2} + 2 \times 2^{3} + \dots + {(n - 1)}^{2 n} + n {.2}^{n + 1} = 2 [2^{n + 10} + 2] \to (2) \\ (1) - (2) \\ 2^{1} + 2^{2} + 2^{3} + 2^{4} + \dots + 2^{n} - n {.2}^{n + 1} = 2^{n + 10} + 2 - 2^{n + 11} - 4 \\ 2^{1} + 2^{2} + 2^{3} + 2^{4} + \dots + 2^{n} = 2^{n + 10} + 2 - 2^{n + 10} .2 - 4 + n {.2}^{n + 1} \\ \Rightarrow \frac{2 (2^{n} - 1)}{2 - 1} = 2^{n + 10} (1 - 2) - 2 + n {.2}^{n + 1} \end{array}$

$\begin{array}{l} \Rightarrow \frac{2 (2^{n} - 1)}{2 - 1} = n {.2}^{n + 1} - 2^{n + 10} - 2 \\ ∴ 2^{n + 10} - 2 = n {.2}^{n + 1} - 2^{n + 10} - 2 \\ \Rightarrow 2^{n + 10} = 2^{n + 1} (n - 1) \\ \Rightarrow 2^{9} = n - 1 \\ \Rightarrow n = 513 \end{array}$