jmc

Question

The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers x, y, and r with |x|>|y|, (x+y)^r=x^r+rx^r-1y^1+r(r-1)2x^r-2y^2+r(r-1)(r-2)3!x^r-3y^3+ are the first three digits to the right of the decimal point in the decimal representation of (10^2002+1)^10/7?

Accepted Answer

By the formula given in the problem, (10^2002 + 1)^10/7 = 10^2860 + 107 10^858 + 107 372 10^-1144 + .The only term which affect the first few digits to the right of the decimal point in this number is 107 10^858 = 10^859 17 = 10^859 0.142857142857 .When 859 is divided by 6, the remainder is 1, so the portion after the decimal point is 0.428571 . Hence, the first three digits are 428.