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Question
Find the coefficient of x50 after simplifying and collecting the like terms in the expansion of (1 + x)1000 + x(1 + x)999 + x2(1 + x)998 + ... + x1000 .
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Solution
Since the above series is a geometric series with the common ratio `x/(1 + x)`
Its sum is `((1 + x)^100 1 - x^1000/(1 + x))/(1 - x/(1 + x))`
= `((1 + x)^1000 - (x^1001)/(1 + x))/((1 + x - x)/(1 + x))`
= `(1 + x)^1001 - x^1001`
Hence, coefficient of x50 is given by
1001C50 = `1001/((50)(951)`
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