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Q.

The sum of the series ∑r=0n (−1)r nCr12r+3r22r+7r23r+15r24r+…+m terms is

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a

2mn−12mn2n−1

b

2mn−12n−1

c

2mn+12n+1

d

None of these

answer is A.

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Detailed Solution

∑nn (−1)rnCr12r+3r22r+7r23r+… upto m terms =∑r=0n (−1)r nCr⋅12r+∑r=0n (−1)r⋅nCr3r22r+∑r=0n (−1)rnCr7r23r+…=1−12n+1−34n+1−78n+…up to M terms=12n+122n+123n… upto m terms =12n1−12nm1−12n=2mn−12mn2n−1

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