Phân số cuối lộ hàng nhé :v
\(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{49\cdot50}\)
\(=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{49\cdot50}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{49}-\dfrac{1}{50}\)
\(=1-\dfrac{1}{50}=\dfrac{49}{50}\)
Đặt \(A=\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+...+\dfrac{1}{49.50}\)
\(A=\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+...+\dfrac{1}{49.50}\)
Ta thấy: \(\dfrac{1}{1.2}=1-\dfrac{1}{2}\)
\(\dfrac{1}{2.3}=\dfrac{1}{2}-\dfrac{1}{3}\)
\(...\)
\(\dfrac{1}{49.50}=\dfrac{1}{49}-\dfrac{1}{50}\)
\(\Rightarrow A=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{49}-\dfrac{1}{50}\)
\(\Rightarrow A=1-\dfrac{1}{50}\)
\(\Rightarrow A=\dfrac{49}{50}\)