Giải thích các bước giải:
Ta có:
$A=\dfrac{3}{5^3}+\dfrac{4}{5^4}+\dfrac{5}{5^5}+...+\dfrac{102}{5^{102}}$
$\to 5A=\dfrac{3}{5^2}+\dfrac{4}{5^3}+\dfrac{5}{5^4}+...+\dfrac{102}{5^{101}}$
$\to 5A-A=\dfrac{3}{5^2}+\dfrac{1}{5^3}+\dfrac{1}{5^4}+...+\dfrac{1}{5^{101}}-\dfrac{102}{5^{102}}$
$\to 4A=\dfrac{3}{5^2}+\dfrac{1}{5^3}+\dfrac{1}{5^4}+...+\dfrac{1}{5^{101}}-\dfrac{102}{5^{102}}$
$\to 4A\cdot 5=\dfrac{3}{5}+\dfrac{1}{5^2}+\dfrac{1}{5^3}+...+\dfrac{1}{5^{100}}-\dfrac{102}{5^{101}}$
$\to 4A\cdot 5-4A=\dfrac{3}{5}-\dfrac{2}{5^2}-\dfrac{1}{5^{101}}-\dfrac{102}{5^{102}}$
$\to 4A\cdot 5-4A<\dfrac{3}{5}-\dfrac{2}{5^2}$
$\to 16A<\dfrac{13}{25}$
$\to A<\dfrac{13}{400}$
