Question

Tính nhanh\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{37.38.39}\)

Answer 1

\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{37.38.39}\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{37.38}-\frac{1}{38.39}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{38.39}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{1482}\right)\)

\(=\frac{1}{2}.\left(\frac{741}{1482}-\frac{1}{1482}\right)\)

\(=\frac{1}{2}.\frac{740}{1482}\)

\(=\frac{185}{741}\)

Chúc bạn học tốt !!! 

Answer 2

Đặt 1/1.2.3 + 1/2.3.4 + ...+ 1/37.38.39 = A

Ta có : 2A = 2/1.2.3 + 2/2.3.4 +...+ 2/37.38.39

         2A = 1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 + ...+ 1/37.38 - 1/38.39

         2A = 1/1.2 - 1/38.39

         2A = 740/1482 = 370/741

           A= 370/741 . 1/2 =........

Answer 3

* Công thức : 

\(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}\right)=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{6}\right)=\frac{1}{2}.\frac{2}{6}=\frac{1}{6}=\frac{1}{1.2.3}\)

( VD cho dễ hiểu ) 

Answer 4

Đặt \(C=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{37.38.39}\)

\(\Rightarrow2C=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+....+\frac{1}{37.38}-\frac{1}{38.39}\)

            \(=\frac{1}{1.2}-\frac{1}{38.39}\)

             \(=\frac{370}{741}\)

\(\Rightarrow C=\frac{370}{741}:2=\frac{185}{741}\)

Answer 5

=185/741

Answer 6

1/2(1/1*2*3+1/2*3*4+...+1/37*38*39)

1/2(1/1*2-1/2*3+1/2*3-1/3*4+...+1/37*38-1/38*39)

1/2(1/1*2-1/38*39)

1/2*370/741

185/741

Answer 7

\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{37.38.39}\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{37.38}-\frac{1}{38.39}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{38.39}\right)\)

\(=\frac{1}{2}\cdot\left(\frac{1}{2}-\frac{1}{1482}\right)\)

\(=\frac{1}{2}\cdot\frac{370}{741}\)

\(=\frac{185}{741}\)