# 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 . - Mathematics

Sum

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 .

#### 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)

Concept: Binomial Theorem for Positive Integral Indices
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#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 8 Binomial Theorem
Solved Examples | Q 14 | Page 137
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