For each positive integer n, let A(n) be the number of digits in the binary representation of n, and let B(n) be the number of ones in the binary representation of n. What is the value of S:

S = (1/2)^[A(1)+B(1)] + (1/2)^[A(2)+B(2)] + (1/2)^[A(3)+B(3)] + ...

This is a great problem. I'd like to see more problems like this on the website.

I solved it correctly using inductive reasoning, but then when I read the other solution methods posted, I was a little disappointed in my approach. There are several very elegant solution methods here -- kudos!

Posted by Rebecca Jaffe on 2005-08-30 03:16:20