Question

Cho S = \(\dfrac{1}{4^2}+\dfrac{1}{5^2}+\dfrac{1}{6^2}+.......+\dfrac{1}{50^2}\) Chứng minh S không là số nguyên

Answer 1

\(S=\dfrac{1}{4^2}+\dfrac{1}{5^2}+...+\dfrac{1}{50^2}< \dfrac{1}{3.4}+\dfrac{1}{4.5}+...+\dfrac{1}{49.50}\)

\(=\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{49}-\dfrac{1}{50}\)

\(=\dfrac{1}{3}-\dfrac{1}{50}=\dfrac{47}{150}\) (1)\(S=\dfrac{1}{4^2}+\dfrac{1}{5^2}...+\dfrac{1}{50^2}>\dfrac{1}{4.5}+\dfrac{1}{5.6}+...+\dfrac{1}{50.51}\)

\(=\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+...+\dfrac{1}{50}-\dfrac{1}{51}\)

\(=\dfrac{1}{4}-\dfrac{1}{51}=\dfrac{47}{204}\) (2)

Từ (1), (2) \(\Rightarrow\dfrac{47}{150}>S>\dfrac{47}{204}\)

\(\Rightarrow S\notin Z\)

\(\Rightarrowđpcm\)