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Question

The sum to (n+1) terms of the following series C02C13+C24C35+ is

A
1n+1
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B
1n+2
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C
1n(n+1)
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D
None of these
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Solution

The correct option is D None of these
x(1x)n=C0xC1x2+C2x3C3x4+
10x(1x)ndx=10(C0xC1x2+C2x3)dx(i)
The integral on the LHS
=01(1t)tn(dt),byputting1x=t
=10(tntn+1)dt=1n+11n+2
whereas the integral on the RHS of (i)
=[C0x22C1x33+C2x44]=C02C13+C24
C02C13+C24 to (n+1) terms
=1n+11n+2=1(n+1)(n+2)

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