12th Putnam 1952

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Problem A5

ai are reals ≠ 1. Let bn = 1 - an. Show that a1 + a2b1 + a3b1b2 + a4b1b2b3 + ... + anb1b2 ... bn-1 = 1 - b1b2 ... bn.

 

Solution

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

 


 

12th Putnam 1952

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