12th Putnam 1952

a_i are reals ≠ 1. Let b_n = 1 - a_n. Show that a₁ + a₂b₁ + a₃b₁b₂ + a₄b₁b₂b₃ + ... + a_nb₁b₂ ... b_n-1 = 1 - b₁b₂ ... b_n.

Solution

Induction on n. Obvious for n = 1. Suppose true for n. Then a₁ + a₂b₁ + a₃b₁b₂ + a₄b₁b₂b₃ + ... + a_n+1b₁b₂ ... b_n = 1 - b₁b₂ ... b_n + a_n+1b₁b₂ ... b_n = 1 - b₁b₂ ... b_n(1 - a_n+1) = 1 - b₁ ... b_n+1, which is the result for n+1.

12th Putnam 1952

© John Scholes
jscholes@kalva.demon.co.uk
5 Mar 2002