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 .

Answer 1

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)`