\(A=\dfrac{3}{5^3}+\dfrac{4}{5^4}+...+\dfrac{103}{5^{103}}\)
\(5A=\dfrac{3}{5^2}+\dfrac{4}{5^3}+...+\dfrac{103}{5^{102}}\)
\(5A-A=\dfrac{3}{5^2}+\dfrac{4}{5^3}-\dfrac{3}{5^3}+\dfrac{5}{5^4}-\dfrac{4}{5^4}+...+\dfrac{103}{5^{102}}-\dfrac{102}{5^{102}}-\dfrac{103}{5^{103}}\)
\(4A=\dfrac{3}{5^2}+\dfrac{1}{5^3}+\dfrac{1}{5^4}+....+\dfrac{1}{5^{102}}-\dfrac{103}{5^{103}}\)
\(4A=\dfrac{3}{5^2}-\dfrac{103}{5^{103}}+\left(\dfrac{1}{5^3}+\dfrac{1}{5^4}+...+\dfrac{1}{5^{102}}\right)\)
Đặt \(B=\dfrac{1}{5^3}+\dfrac{1}{5^4}+...+\dfrac{1}{5^{102}}\)
\(5B=\dfrac{1}{5^2}+\dfrac{1}{5^3}+...+\dfrac{1}{5^{101}}\)
\(5B-B=\dfrac{1}{5^2}-\dfrac{1}{5^{102}}\)
\(4B=\dfrac{1}{5^2}-\dfrac{1}{5^{102}}\)
\(B=\dfrac{1}{4}.\left(\dfrac{1}{5^2}-\dfrac{1}{5^{102}}\right)\)
\(4A=\dfrac{3}{5^2}-\dfrac{103}{5^{103}}+\dfrac{1}{4}.\left(\dfrac{1}{5^2}-\dfrac{1}{5^{102}}\right)\)
\(4A=\dfrac{3}{5^2}+\dfrac{1}{4.5^2}-\left(\dfrac{103}{5^{103}}+\dfrac{1}{4.5^{102}}\right)\)
\(4A< \dfrac{3}{5^2}+\dfrac{1}{4.5^2}\)
\(4A< \dfrac{13}{100}\)
\(A< \dfrac{13}{400}\)
