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Question

# The sum to (n+1) terms of the series C02−C13+C24−C35+⋯ is

A
1n+1
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B
1n+2
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C
1n(n+1)
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D
1(n+1)(n+2)
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Solution

## The correct option is D 1(n+1)(n+2)Given:C02−C13+C24−⋯+(−1)nCnn+2Consider the expansion of (1−x)n=C0−C1x+C2x2−⋯+(−1)n CnxnMultiply both sides with x, we getx(1−x)n=C0x−C1x2+C2x3−⋯+(−1)n Cnxn+1Integrating both sides w.r.t x from 0 to 1, ∫10x(1−x)n dx=∫10(C0x−C1x2+C2x3−⋯+(−1)n Cnxn+1)dx⇒[(1−x)n+2n+2−(1−x)n+1n+1]10=[C0x22−C1x33+C2x44−⋯+(−1)n Cnxn+2n+2]10⇒1(n+1)(n+2)=C02−C13+C24−⋯+(−1)nCnn+2Hence, option D.

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