please write the question more clearly.

are you sore it's not from n=1 to n=100? it would be more easier

*n=0

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

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please write the question more clearly.

are you sore it's not from n=1 to n=100? it would be more easier

*n=0

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

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n=1 => 2^1=2

n=2 => 2^2=4

n=3 => 2^3=8

n=4 => 2^4=16

...............

n=99=> 2^99

n=100 => 2^100

So

2+4+8+16+....+2^100

But we know that if we multiply the number with 1 the result will be the same

So the ecuation will be:

(2-1)(2+4+8+16+....+2^99+2^100)=

(2-1)(2^1+2^2+2^3+....+2^99+2^100)=

when we multiply we add the powers so 2^3 for example comes from 2*2^2=2^(2+1)=2^3

(2-1)(2^1+2^2+2^3+....+2^99+2^100)=

2^2+2^3+2^4+...................+2^100+2^101 -

2^1-2^2-2^3-2^4-..............-2^99-2^100

so remains 2^1+2^101=

2+2^101=

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

- - - - -

-------------------------------------------------------------------------------------------------

-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