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The value of  [C0 / (1 ∙ 3)] – [C1 / (2 ∙ 3)] + [C2 / (3 ∙ 3)] – [C3 / (4 ∙ 3)] + ….. [{(–1)n Cn} / {(n + 1) ∙ 3}] is____

(a) [1 / {3(n + 1)}]   

(b) [1 / (n + 1)] 

(c) [3 / (n + 1)] 

(d) none of these

1 Answer

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Best answer

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)}]

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