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

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510

511

512

513

answer is D.

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Detailed Solution

We have 2n+10=2×22+3×23+4×24+…+n×2n⇒ 22n+10=2×23+3×24+…+(n−1)×2n+ n×2n+1 Subtracting, we get −2n+10=2⋅22+23+24+…+2n−n⋅2n+1 =8+82n−2−12−1−n×2n+1 =2n+1−(n)2n+1⇒ 210=2n−2 ⇒ n=513

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