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Question

If (1+x)n=C0+C1x+C2x2+.......+Cnxn. then C0+2C1+3C2+.........+(n+1)Cn is equal to-

A
2n1(n+2)
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B
2n(n+1)
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C
2n1(n+1)
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D
2n(n+2)
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Solution

The correct option is A 2n−1(n+2)(Hn)n+C0+C1x+C2x2+........CnxnC0+2C1+3C2+.........+(n+1)Cn=2(Hn)n=C0+C1x+C2x2+............Cnxn−−−−−(1)n(1+x)n−1=0+C1+2C2x+3C3x2+.....+xCnxn−1Put x=1n(2)n−1=C1+2C2+3C3+.......+nCn−−−−−−−−(2)Put x=1 in (1)2n=C0+C1+C2+C3+.......Cn−−−−−−−−−(3)(2)+(3) & C0+2C1+3C2+..........+(n+1)Cn=n2n−1+2n=2n[1+n2]C0+2C1+3C2+........(n+1)Cn=2n+1[2+n]

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