tính nhanh 3/1+3/1+2+3/1+2+3+…+3/1+2+3+…+10 1/1+2+1/1+2+3+1/1+2+3+4+…+1/1+2+3+4+…..+50

1 bình luận về “tính nhanh 3/1+3/1+2+3/1+2+3+…+3/1+2+3+…+10 1/1+2+1/1+2+3+1/1+2+3+4+…+1/1+2+3+4+…..+50”

1. \(A=3+\frac{3}{1+2}+\frac{3}{1+2+3}+…+\frac{3}{1+2+3+4+…+100}\)

   \(A=3\left(1+\frac{1}{1+2}+\frac{1}{1+2+3}+…+\frac{1}{1+2+3+4+…+100}\right)\)

   \(B=1+\frac{1}{1+2}+\frac{1}{1+2+3}+…+\frac{1}{1+2+3+4+…+100}\)

   \(B=1+\frac{1}{1+2}+\frac{1}{1+2+3}+…+\frac{1}{1+2+3+4+…+100}\)

   \(B=1+\frac{1}{\left(1+2\right).2:2}+\frac{1}{\left(1+3\right).3:2}+\frac{1}{\left(1+4\right).4:2}+…+\frac{1}{\left(1+100\right).100:2}\)

   \(B=\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+…+\frac{2}{100.101}\)

   \(B=2\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+…+\frac{1}{100.101}\right)\)

   \(B=2\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+…+\frac{1}{100}-\frac{1}{101}\right)\)

   \(B=2.\left(1-\frac{1}{101}\right)\)

   \(B=2.\frac{100}{101}=\frac{200}{101}\)

   Ta có:

   A=3.B

   Rightarrow $A=3.\frac{200}{101}=\frac{600}{101}$

    \(A=\frac{600}{101}\)

   Trả lời