What is the 1000th digit to the right of the decimal point in the decimal representation of (1+√2)^3000?

This problem can be solved by algebra alone, without the need for computers or calculators

(In reply to re: Solution [FK's] by Tristan)

"Newton's Formula" used by FK is more commonly called "The Binomial Theorem". Expanding each of (1+sqrt(2))^n and (1-sqrt(2))^n and adding the expansions will give a whole number result because the terms involving the odd powers of sqrt(2) cancel out and each  term involving an even power of sqrt(2) is an integer equal to twice an even power of sqrt(2) times a binomial coefficient.

Newton generalized the ordinary binomial theorem that holds for positive integer powers to include negative and rational powers, and this is perhaps why the binomial theorem is sometimes called Newton's Theorem, or Newton's Binomial Theorem.
 

Edited on April 28, 2005, 8:18 pm

Posted by Richard on 2005-04-28 18:26:07