Question

\(S=\dfrac{1}{5^2}+\dfrac{1}{5^4}+...+\dfrac{1}{5^{2022}}.CMR:S< \dfrac{1}{24}\)

Answer 1

\(S=\dfrac{1}{5^2}+\dfrac{1}{5^4}+...+\dfrac{1}{5^{2022}}\)

=>\(25\cdot S=1+\dfrac{1}{5^2}+...+\dfrac{1}{5^{2020}}\)

=>\(25S-S=1+\dfrac{1}{5^2}+...+\dfrac{1}{5^{2020}}-\dfrac{1}{5^2}-\dfrac{1}{5^4}-...-\dfrac{1}{5^{2022}}\)

=>\(24S=1-\dfrac{1}{5^{2022}}\)

=>\(S=\dfrac{1}{24}-\dfrac{1}{24\cdot5^{2022}}< \dfrac{1}{24}\)