Prove that (1+x)(1+x2)(1+x4) ... (1+x2n-1) = 1 + x + x2 + x3 + ... + x2n-1.
Solution
Multiply the lhs by 1-x. We have (1-x)(1+x) = 1-x2, (1-x2)(1+x2) = 1-x4, (1-x4)(1+x4) = 1-x8, and so on. So (1-x)lhs = 1-x2n. Similarly, (1-x)rhs = 1-x2n.
© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03