Question

[thị phương linh lê]

☹➞cac ban giup minh cau nay voi 

☛ tinh tong 

M = 1+\(\dfrac{1}{5}\)+\(\dfrac{3}{35}\)+...\(\dfrac{3}{9603}\)+\(\dfrac{3}{9999}\)

➜giup minh nhanh nhe minh dang can gap lam moi nguoi a 

➞thank you cac ban ❕

Answer 1

[Thuỳ Linh Nguyễn]

\(M=1+\dfrac{1}{5}+\dfrac{3}{35}+...+\dfrac{3}{9999}\\ =\dfrac{3}{3}+\dfrac{3}{15}+\dfrac{3}{35}+...+\dfrac{3}{9999}\\ =\dfrac{3}{2}\left(\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+...+\dfrac{2}{99\cdot101}\right)\\ =\dfrac{3}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}\right)\\ =\dfrac{3}{2}\left(1-\dfrac{1}{101}\right)=\dfrac{3}{2}\cdot\dfrac{100}{101}=\dfrac{150}{101}\)