Question

tính nhanh:

\(\frac{1}{13}\)+\(\frac{3}{13.23}\)+\(\frac{3}{23.33}\)+...+\(\frac{3}{1993.2003}\)

Giúp mik nhoa, mik đang cần gấp lắm ^-^

Answer 1

\(\frac{1}{13}+\frac{3}{13\cdot23}+\frac{3}{23\cdot33}+...+\frac{3}{1993\cdot2003}\)

\(=\frac{1}{13}+\left[\frac{3}{13\cdot23}+\frac{3}{23\cdot33}+...+\frac{3}{1993\cdot2003}\right]\)

\(=\frac{1}{13}+\left[\frac{3}{10}\left[\frac{1}{13\cdot23}+\frac{1}{23\cdot33}+...+\frac{1}{1993\cdot2003}\right]\right]\)

\(=\frac{1}{13}+\left[\frac{3}{10}\left[\frac{1}{13}-\frac{1}{23}+\frac{1}{23}-\frac{1}{33}+...+\frac{1}{1993}-\frac{1}{2003}\right]\right]\)

\(=\frac{1}{13}+\left[\frac{3}{10}\left[\frac{1}{13}-\frac{1}{2003}\right]\right]\)

\(=\frac{1}{13}+\left[\frac{3}{10}\cdot\frac{1990}{26039}\right]\)

\(=\frac{1}{13}+\frac{597}{26039}\)

\(=\frac{200}{2003}\)

Answer 2

Đặt A= 1/13 + 3/13.23 + 3/ 23.33 + ... + 3/1993.2003 

A- 1/13 = 3/13.23 + 3/ 23.33 + ... + 3/1993.2003 

10/3 ( A-1/3) =  10/3. (3/13.23 + 3/ 23.33 + ... + 3/1993.2003) 

10/3A - 10/9 = 10/13.23 + 10/ 23.33 + ... + 10/1993.2003 

10/3A - 10/9  = 1/13 - 1/23 + 1/23 - 1/33 +...+ 1/1993- 1/2003

10/3A = 1/13 - 1/2003 + 10/9

10/3 A= ? 

đến đây bn tự làm nha

10/3A - 10/9 = 1/13 

Answer 3

Ta loại \(\frac{1}{13}\)ra khỏi biểu thức để dễ tính hơn, sau đó cộng vào.

Gọi A là biểu thức \(\frac{3}{13.23}+\frac{3}{23.33}+...+\frac{3}{1993.2003}\)

Gấp A lên \(\frac{10}{3}\)lần, ta có :

\(A\times\frac{10}{3}=\frac{10}{3}\times\left(\frac{3}{13.23}+\frac{3}{23.33}+...+\frac{3}{1993.2003}\right)\)

\(A\times\frac{10}{3}=\frac{10}{13.23}+\frac{10}{23.33}+...+\frac{10}{1993.2003}\)

\(A\times\frac{10}{3}=\frac{1}{13}-\frac{1}{23}+\frac{1}{23}-\frac{1}{33}+...+\frac{1}{1993}-\frac{1}{2003}\)

\(A\times\frac{10}{3}=\frac{1}{13}-\frac{1}{2003}\)

\(A\times\frac{10}{3}=\frac{1990}{26039}\)

\(A=\frac{1990}{26039}\div\frac{10}{3}\)

\(A=\frac{597}{26039}\)

Biểu thức trên = \(\frac{597}{26039}+\frac{1}{13}=\frac{200}{2003}\)

Answer 4

\(\frac{200}{2003}\) nha 

Nguyễn Linh Ngọc