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Question

Find the sum of the series
r=0n(1)r nCr[12r+3r22r+7r23r+15r24rupto m terms]


A

2n12mn(2n1)

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B

2m12n(2n1)

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C

2mn12mn(2n1)

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D

2mn12mn(2m1)

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Solution

The correct option is C

2mn12mn(2n1)


=nr=0(1)r nCr(12)r+nr=0(r)r nCr(34)r+nr=0(1)r nCr(78)r+ upto m terms
=(112)n+(134)n+(178)n+ upto m terms
[using nr=0(1)r nCr.xr=(1x)n]
=(12)n+(14)n+(18)n+ upto m terms
=(12)n[1(12n)m112n]=2mn12mn(2n1)


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