$#nobody$

`a)`

`A = 2 + 2^2 + 2^3 + 2^4 + ... + 2^99 + 2^100`

`= (2 + 2^2) + 2^2 (2 + 2^2) + ... + 2^98 (2 + 2^2)`

`= 6 + 2^2 . 6 + ... + 2^98 . 6`

`= 6.(1 + 2^2 + ... + 2^98) \vdots 6(đpcm)`

`A = 2 + 2^2 + 2^3 + 2^4 + ... + 2^99 + 2^100`

`= (2 + 2^2 + 2^3 + 2^4) + ... + 2^96 (2 + 2^2 + 2^3 + 2^4)`

`= 30 + ... + 2^96 . 30`

`= 30 . (1 + .. + 2^96) \vdots 30 (đpcm)`

 `b)`

`B = 5 + 5^2 + 5^3 +  ... + 5^94 + 5^95 + 5^96`

`= (5 + 5^2 + 5^3) + ... + (5^94 + 5^95 + 5^96)`

`= 5.(1 + 5 + 5^2) + ... + 5^94 . (1 + 5 + 5^2)`

`= 5 . 31 + ... + 5^94 . 31`

`= 31 . (5 + .. + 5^94) \vdots 31 (đpcm)`

`B = 5 + 5^2 + 5^3 + 5^4 + ... + 5^93 + 5^94 + 5^95 + 5^96`

`= (5 + 5^2 + 5^3 + 5^4) + ... + (5^93 + 5^94 + 5^95 + 5^96)`

`= 5.(1 + 5 + 5^2 + 5^3) + ... + 5^93 . (1 + 5 + 5^2 + 5^3)`

`= 5 . 6 . 26 + ... + 5^93 . 6 . 26`

`= 6 . 26 . (5 + ... + 5^93) \vdots 6` và `\vdots 26(đpcm)`

`B = 5 + 5^2 + 5^3 + ... + 5^94 + 5^95 + 5^96`

`= (5 + 5^2 + ... + 5^6) + ... + (5^91 + 5^92 + ... + 5^96)`

`= (5 + 5^2 + ... + 5^6) + ... + 5^90 . (5 + 5^2 + ... + 5^6)`

`= (5 + 5^2 + ... + 5^6) . (1 + ... + 5^90)`

`= 155 . 126 . (1 + ... + 5^90) \vdots 126 (đpcm)`

`c)`

`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 = 2^101 - 2`

`B = 5 + 5^2 + 5^3 + ... + 5^96`

`5B = 5(5 + 5^2 + 5^3 + ... + 5^96)`

`5B = 5^2 + 5^3 + 5^4 + ... + 5^97`

`5B - B = (5^2 + 5^3 + 5^4 + ... + 5^97) - (5 + 5^2 + 5^3 + ... + 5^96)`

`4B = 5^97 - 5`

`B = (5^97 - 5)/4`

`C = 2^100 - 2^99 + 2^98 - 2^97 + ... + 2^2 - 2`

`C = 2^99 . (2 - 1) + 2^97 . (2 - 1) + ... + 2.(2 - 1)`

`C = 2^99 + 2^97 + ... + 2`

`4C = 2^2 . (2^99 + 2^97 + ... + 2)`

`4C = 2^101 + 2^99 + ... + 2^3`

`4C - C = (2^101 + 2^99 + ... + 2^3) - (2^99 + 2^97 + ... + 2)`

`3C = 2^101 - 2`

`C = (2^101 - 2)/3`