Question

In the polynomial `(x-1)(x-2)(x-3)cdots(x-100)`. Find the coefficient of `x^99` is?

Answer 1

`a_99 = 5050`

Explanation:

Calling `p_n(x) = prod_(k=1)^n(x-k)` we have

we now that their roots are `1,2,3,cdots,n`

and the coefficients for

`p_n(x) = a_0+a_1x+a_2x^2+ cdots + a_(n-1) x^(n-1) + x^n`

are liked to the roots values - For instance

`a_0 = n!, a_(n-1) = sum_(k=1)^n k = (n(n+1))/2`

so `a_99 = (100(100+1))/2 = 5050`