Question

(1+2+2^2+.....+2^99)÷(2^100-1)=​

Answer 1

$(1+2+2^2+...+2^{99}):(2^{100}-1)=\,?\\ \\\\S = 1+2+2^2+...+2^{99}\\2S = 2+2^2+2^3+...+2^{99}+2^{100}\\ 2S = 1+2+2^2+...+2^{99}+2^{100}-1\\ 2S = S+2^{100}-1\\ S = 2^{100}-1\\ \\\\ (1+2+2^2+...+2^{99}):(2^{100}-1)=\\ \\ =(2^{100}-1):(2^{100}-1)\\ \\ =\boxed{1}$