Question

A 2+2²+....+2¹⁹+2²⁰

B 2+2²+....+2⁹⁹+2¹⁰⁰

C 3+3²+.....+3⁹+3¹⁰

Answer 1

a: 2A=2^2+2^3+...+2^21

=>A=2^21-2

b: B=2+2^2+...+2^100

=>2B=2^2+2^3+...+2^101

=>B=2^101-2

c: C=3+3^2+...+3^10

=>3C=3^2+3^3+...+3^11

=>2C=3^11-3

=>C=(3^11-3)/2

Answer 2

`A = 2 + 2^2 + ... + 2^20`

`=> 2A = 2^2 + 2^3 + ... +2^21`

`=> 2A-A = (2^2 + 2^3 + ... + 2^21) - (2 + 2^2 + ...  +2^20)`

`=> A      = 2^21 - 2`

`B = 2 + 2^2 + ... + 2^99 + 2^100`

`=>2B = 2^2 + 2^3 + ... + 2^100 + 2^101`

`=> 2B-B = (2^2 + 2^3 + ... + 2^101)- (2 + 2^2 + ... + 2^100)`

`=> B      = 2^101 - 2`

`C = 3 + 3^2 +  .... + 3^10`

`=>3C = 3^2 + 3^3 + ... +3^11`

`=>3C - C = (3^2 + 3^3 + ... +3^11) - (3 + 3^2 +  .... + 3^10)`

`=> 2C     = 3^11 - 3`

`=>  C      = (3^11 - 3)/2