Question bank/JEE Main/Mathematics
JEE MainMathematics
The positive integral value of n such that
1·21 + 2·22 + 3·23 + 4·24 +...........+ n·2n = 2 + 2n+5, is
Show answer and solution
Let Sn = 1·21 + 2·22 + 3·23 + 4·24 +...........+ n·2n.......... (1)
⇒ 2.Sn = 1·22 + 2·23 +......................+ (n - 1)2n +
n·2n+1.......... (2)
∴ On subtracting, we get
⇒ -Sn = (21 + 22 + 23 +.........+2n) - n·2n+1 = 2(2n - 1) -
n·2n+1
∴ Sn = n·2n+1 - 2(2n - 1) = n·2n+1 - 2n+1 + 2 = 2 + (n - 1)2n+1
But 2 + (n - 1) 2n+1 = 2 + 2n+5 (Given)
⇒ n - 1 = 24
∴ n = 17
⇒ 2.Sn = 1·22 + 2·23 +......................+ (n - 1)2n +
n·2n+1.......... (2)
∴ On subtracting, we get
⇒ -Sn = (21 + 22 + 23 +.........+2n) - n·2n+1 = 2(2n - 1) -
n·2n+1
∴ Sn = n·2n+1 - 2(2n - 1) = n·2n+1 - 2n+1 + 2 = 2 + (n - 1)2n+1
But 2 + (n - 1) 2n+1 = 2 + 2n+5 (Given)
⇒ n - 1 = 24
∴ n = 17
Source: Grass-G / M-01 Mathematics Paper-1.docx
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