\(M=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{50^2}\)
\(M< \dfrac{1}{4}+\left(\dfrac{1}{2.3}+...+\dfrac{1}{49.50}\right)=\dfrac{1}{4}+M_1\)
\(M_1=\left(\dfrac{1}{2}-\dfrac{1}{3}\right)+\left(\dfrac{1}{3}-\dfrac{1}{4}\right)...+\left(\dfrac{1}{48}-\dfrac{1}{49}\right)+\left(\dfrac{1}{49}-\dfrac{1}{50}\right)\)
\(M_1=\dfrac{1}{2}+\left(-\dfrac{1}{3}+\dfrac{1}{3}\right)+...+\left(-\dfrac{1}{49}+\dfrac{1}{49}\right)-\dfrac{1}{50}=\dfrac{1}{2}-\dfrac{1}{50}\)
\(M< \dfrac{1}{4}+\dfrac{1}{2}-\dfrac{1}{50}=\dfrac{3}{4}-\dfrac{1}{50}< \dfrac{3}{4}=>dpcm\)
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