A = $\frac{1}{3}$ + $\frac{2}{3^{2}}$ + $\frac{3}{3^{3}}$ + ... + $\frac{102}{3^{102}}$
$\frac{1}{3}$ A = $\frac{1}{3^{2}}$ + $\frac{2}{3^{3}}$ + $\frac{3}{3^{4}}$ + ... + $\frac{102}{3^{103}}$
A - $\frac{1}{3}$ A = ($\frac{1}{3}$ + $\frac{2}{3^{2}}$ + $\frac{3}{3^{3}}$ + ... + $\frac{102}{3^{102}}$) - ($\frac{1}{3^{2}}$ + $\frac{2}{3^{3}}$ + $\frac{3}{3^{4}}$ + ... + $\frac{102}{3^{103}}$)
$\frac{2}{3}$ A = $\frac{1}{3}$ + $\frac{1}{3^{2}}$ + $\frac{1}{3^{3}}$ + ... + $\frac{1}{3^{102}}$ - $\frac{102}{3^{103}}$
A = ($\frac{1}{3}$ + $\frac{1}{3^{2}}$ + $\frac{1}{3^{3}}$ + ... + $\frac{1}{3^{102}}$ - $\frac{102}{3^{103}}$) · $\frac{3}{2}$
Ta có: A = ($\frac{1}{3}$ + $\frac{1}{3^{2}}$ + $\frac{1}{3^{3}}$ + ... + $\frac{1}{3^{102}}$ - $\frac{102}{3^{103}}$) · $\frac{3}{2}$ < ($\frac{1}{3}$ + $\frac{1}{3^{2}}$ + $\frac{1}{3^{3}}$ + ... + $\frac{1}{3^{102}}$) · $\frac{3}{2}$
Đặt B = ($\frac{1}{3}$ + $\frac{1}{3^{2}}$ + $\frac{1}{3^{3}}$ + ... + $\frac{1}{3^{102}}$) · $\frac{3}{2}$
Ta có:
B = ($\frac{1}{3}$ + $\frac{1}{3^{2}}$ + $\frac{1}{3^{3}}$ + ... + $\frac{1}{3^{102}}$) · $\frac{3}{2}$
$\frac{1}{3}$ B = ($\frac{1}{3^{2}}$ + $\frac{1}{3^{3}}$ + ... + $\frac{1}{3^{103}}$) · $\frac{3}{2}$
B - $\frac{1}{3}$ B = ($\frac{1}{3}$ + $\frac{1}{3^{2}}$ + $\frac{1}{3^{3}}$ + ... + $\frac{1}{3^{102}}$) · $\frac{3}{2}$ - ($\frac{1}{3^{2}}$ + $\frac{1}{3^{3}}$ + ... + $\frac{1}{3^{103}}$) · $\frac{3}{2}$
$\frac{2}{3}$ B = $\frac{3}{2}$ · [($\frac{1}{3}$ + $\frac{1}{3^{2}}$ + $\frac{1}{3^{3}}$ + ... + $\frac{1}{3^{102}}$) - ($\frac{1}{3^{2}}$ + $\frac{1}{3^{3}}$ + ... + $\frac{1}{3^{103}}$)]
B = $\frac{9}{4}$ · [$\frac{1}{3}$ - $\frac{1}{3^{103}}$]
B = $\frac{9}{4}$ · $\frac{3^{102} - 1}{3^{103}}$
B = $\frac{1}{4}$ · $\frac{3^{102} - 1}{3^{101}}$ < $\frac{1}{4}$ · $\frac{3^{102}}{3^{101}}$
Đặt C = $\frac{1}{4}$ · $\frac{3^{102}}{3^{101}}$
Ta có:
C = $\frac{1}{4}$ · 3
C = $\frac{3}{4}$
⇒ A < B < C = $\frac{3}{4}$
Vậy A < $\frac{3}{4}$
