Question

帮帮忙（1-1）/2^1+（2-1）/2^2+(3-1)/2^3+...+(100-1)/2^100 这道题的过程是什么

Answer 1

(1-1)/2^1+(2-1)/2^2+(3-1)/2^3+...+(100-1)/2^100

=0/2^1+1/2^2+2/2^3+.+99/2^100

=1/2^2+2/2^3+.+99/2^100

令s=1/2^2+2/2^3+.+99/2^100

s/2=1/2^3+2/2^4+.+99/2^101

s-s/2=1/2^2+1/2^3+.+1/2^100-99/2^101

s/2=1/4*[1-(1/2)^99]/(1-1/2)-99/2^101

s/2=1/2*[1-(1/2)^99]-99/2^101

s/2=1/2-1/2^100-99/2^101

s/2=1/2-2*1/2^101-99/2^101

s/2=1/2-101/2^101

s=1-101/2^100

即(1-1)/2^1+(2-1)/2^2+(3-1)/2^3+...+(100-1)/2^100=1-101/2^100