Ta đặt
$A = 1 + 2^2 + \cdots + 2^{101}$
Khi đó
$2A = 2 + 2^2 + \cdots + 2^{102}$
Vậy
$A = 2A - A = (2 + 2^2 + \cdots + 2^{102}) - (1 + 2^2 + \cdots + 2^{101})$
$= 2^{102} - 1$
Ta sẽ so sánh $2^{102} - 1$ và $5.2^{100}$
Ta xét
$\dfrac{2^{102}-1}{5.2^{100}} = \dfrac{2^{102}}{5.2^{100}} - \dfrac{1}{5.2^{100}} = \dfrac{4}{5} - \dfrac{1}{5.2^{100}} < \dfrac{4}{5} < 1$
Vậy $2^{102}-1 < 5.2^{100}$