Correct option: (a) [1 / {3(n + 1)}]
Explanation:
∵ (1 + x)n = C0 + C1x + C2x2 + ------- Cnxn
∴ 0∫–1 (1 + x)n = 0∫–1 (C0 + C1x +C2x2 + ----- Cnxn)dx
∴ [{(1 + x)n+1} / (n + 1)]0–1 = [C0x+ {(C1x2) / 2} + {(C2x3) / 3} + ------ {(Cnxn+1) / (n + 1)]0–1
∴ [1 / (n + 1)] = C0 – (C1 / 2) + (C2 / 3) + ------ (– 1)n [Cn / (n + 1)]
∴ The given expression = (1 / 3) [C0 – (C1 / 2) + (C2 / 3) ------]
= (1 / 3) ∙ [1 / (n + 1)] = [1 / {3(n + 1)}]