$\text { Đặt }$ `A = 2 + 2^2 + 2^3 + ... + 2^100`
`⇒ A = (2 + 2^2 + 2^3 + 2^4) + .... + (2^97 + 2^98 + 2^99 + 2^100)`
`⇒ A = 2 . (1 + 2 + 2^2 + 2^3) + .... + 2^97 . (1 + 2 + 2^2 + 2^3)`
`⇒ A = 2 . 15 + ... + 2^97 . 15`
`⇒ A = 2 . 3 . 5 + ... + 2^97 . 3 . 5`
`⇒ A = 5 . (2 . 3 + ... + 2^97 . 3)` $\vdots$ `5 (1)`
$\text { Mặt khác, ta có: }$
`A = 2 + 2^2 + 2^3 + ... + 2^100`
`⇒ A = (2 + 2^2 + 2^3 + 2^4 + 2^5) + ... + (2^96 + 2^97 + 2^98 + 2^99 + 2^100)`
`⇒ A = 2 . (1 + 2 + 2^2 + 2^3 + 2^4) + .... + 2^97 . (1 + 2 + 2^2 + 2^3 + 2^4)`
`⇒ A = 2 . 31 + ... + 2^97 . 31`
`⇒ A = 31 . (2 + ... + 2^97)` $\vdots$ `31 (2)`
`(1), (2) ⇒ đpcm`
