Ta có:

`S = 1 + 2^2 + 2^3 + ... + 2^99`

`2S = 2 * (1 + 2^2 + 2^3 + ... + 2^99)`

`2S = 2 + 2^3 + 2^4 + ... + 2^100`

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

`S = 2 + 2^3 + 2^4 + ... + 2^100 - 1 - 2^2 - 2^3 - ... - 2^99`

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

`S = 1 - 4 + 2^100`

`S = 2^100 - 3`

Ta có: `S = 2^100 - 3`

Vì `3 > 0` nên `2^100 - 3 < 2^100`

Mà `2^100 < 5 * 2^100`

Nên `2^100 - 3 < 5 * 2^100`

Vậy `S < 5 * 2^100`