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Question: The positive integer 'n' for which 2 × 2<sup>2</sup> + 3 × 2<sup>3</sup> + 4 × 2<sup>4</sup> + ………....

The positive integer 'n' for which

2 × 22 + 3 × 23 + 4 × 24 + ……….n × 2n = 2n +10 is :

A

510

B

511

C

512

D

513

Answer

513

Explanation

Solution

2n+10 = 2×22 + 3×23 + 4 × 24 + …..n.2n

2.(2n+10) = 2×23+3×24+….(n–1).2n+n.2n+1

On subtracting we get :–

–2n+10 = 2×22 + 23 + 24 +….25 +….2n – n.2n+1

– 2n+10 = 8 + –n.2n+1

̃ –2n+10 = 8 + 8.2n–2 – 8 – n.2n+1

̃ – 2n+10 = 2n+1 – n. 2n+1

̃ –2n+10 = 2n+1(1– n)

̃ –(2n . 210) = 2n . 2(1 – n)

̃ 210 = 2n – 2

̃ 1024 = 2n – 2

̃ n = 513