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

1 Answer
Oct 23, 2017

#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#