Ques. 1JEE Main 2024, 6 April (Evening Shift)
If S(x) = (1 + x) + 2(1 + x)2 + 3(1 + x)3 + + 60(1 + x)60, x 0, and (60)2 S(60) = a(b)b + b, where a, b N, then (a + b) equal to .
Answer
3660
Solution
AGP with t = 1 + x: S = t + 2t2 + + 60t60, tS = t2 + + 59t60 + 60t61
Subtract: (1 - t)S = t + t2 + + t60 - 60t61 = t (t60 - 1 )t - 1 - 60t61
t - 1 = x: -xS = t (t60 - 1 )x - 60t61
x2S = 60x t61 - t61 + t
Put x = 60, t = 61:
(60)2 S(60) = 3600 6161 - 6161 + 61
= 3599 (61)61 + 61 a = 3599, b = 61
a + b = 3660
Answer: 3660