Question

1*2+2*2^2+3*2^3+4*2^4+.......+100*2^100=?

Answer 1

see this can be done as
  S=1*2 + 2*2^2 + 3*2^3...........100*2^100
2S=         1*2^2 + 2*2^3...........  99*2^100 + 100*2^101
-              -        -                       -             -
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-S=2 + 2^2 + 2^3 +............2^100 -100*2^101
using sum of gp formula we will get
-S=2(2^100-1) -100* 2^101 
S=100*2^101 - 2^101 - 2
S=2^101(99) - 2