$2^{n}$- 1- 2- $2^{2}$- $2^{3}$- ... - $2^{100}$= 1
=> $2^{n}$- (1+2+$2^{2}$+ $2^{3}$+...+ $2^{100}$)= 1 (1)
Đặt A= 1+ 2+ $2^{2}$+ $2^{3}$+ ... + $2^{100}$
=> 2A= 2+ $2^{2}$+ $2^{3}$+ ... + $2^{100}$+ $2^{101}$
=> 2A-A= ( 2+ $2^{2}$+ $2^{3}$+ ... + $2^{100}$+ $2^{101}$) - ( 1+2+$2^{2}$+ $2^{3}$+...+ $2^{100}$)
=> A= $2^{101}$ - 1
(1) => $2^{n}$ - ( $2^{101}$ - 1 )= 1
<=> $2^{n}$ - $2^{101}$ + 1 = 1
<=> $2^{n}$= $2^{101}$ + 1 - 1
<=> $2^{n}$= $2^{101}$
=> n = 101
