2011/018) Prove that in the product

(1-x+x^2-x^3+ ..... - x^99 + x^100) ( 1 +x +x^2 + .. x^99 + x^100) after multiplying there does not appear a term of x odd degree.

this problem is solved by taking coefficient in a couple of books and I try to solve in a different approach

LHS = (1-x^101)/(1-x) * (1+ x^101)/(1+x)
= (1-x^202)/(1-x^2)

if we define f(x) = (1-x^101)/(1-x) then f(x^2) is the given expression and as it is division of p(x^2) by q(x^2) so cannot have any term other than (x^2)^2k or odd power of x