`3)`
`a) A = 2 + 2^2 + 2^3 + ... + 2^100`
`=> 2A = 2(2 + 2^2 + 2^3 + ... + 2^100)`
`=> 2A = 2^2 + 2^3 + 2^4 + ... + 2^101`
`=> 2A - A = (2^2 + 2^3 + 2^4 + ... + 2^101) - (2 + 2^2 + 2^3 + ... + 2^100)`
`=> A = (2^2 + 2^3 + 2^4 + ... + 2^100) + 2^101 - 2 - (2^2 + 2^3 + ... + 2^100)`
`=> A = 2^101 - 1`
`b) B = 1 + 5 + 5^2 + ... + 5^50 + 5^51`
`=> 5B = 5(1 + 5 + 5^2 + ... + 5^50 + 5^51)`
`=> 5B = 5 + 5^2 + 5^3 + ... + 5^51 + 5^52`
`=> 5B - B = (5 + 5^2 + 5^3 + ... + 5^51 + 5^52) - (1 + 5 + 5^2 + ... + 5^50 + 5^51)`
`=> 4B =(5 + 5^2 + 5^3 + ... + 5^51) + 5^52 - 1 - (5 + 5^2 + ... + 5^50 + 5^51)`
`=> 4B = 5^52 - 1`
`=> B = (5^52 - 1)/4`
`4)`
`A = 2 + 2 + 2^2 + 2^3 + ... + 2^99`
`A = 2^2 + 2^2 + 2^3 + ... + 2^99`
`=> 2A = 2(2^2 + 2^2 + 2^3 + ... + 2^99)`
`=> 2A = 2^3 + 2^3 + 2^4 + ... + 2^100`
`=> 2A - A = (2^3 + 2^3 + 2^4 + ... + 2^100) - (2^2 + 2^2 + 2^3 + ... + 2^99)`
`=> A = (2^3 + 2^4 + ... + 2^99) + 2^3 + 2^100 - 2^2 - 2^2 - (2^3 + ... + 2^99)`
`=> A = 2^3 + 2^100 - 2^2 - 2^2`
`=> A = 8 + 2^100 - 4 - 4`
`=> A = 8 + 2^100 - 8`
`=> A = 2^100`
