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Question

The coefficient of x50 in (1+x)1000+x(1+x)999+x2(1+x)998+...+x1000is

A
1001C50
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B
1000C50
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C
1001C51
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D
1000C51
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Solution

The correct option is A 1001C50
Clearly, the given series is a geometric progression in which
a=(1+x)1000,r=x(1+x)999(1+x)1000=x(1+x) and n=1001.
Given sum =a(1rn)(1r)
=(1+x)1000×{1(x1+x)1001}(1x1+x)
=(1+x)1001x1001

Coefficient of x50 in the above expansion = 1001C50

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