Question

[Vũ Đức Long Hải]

Tính nhanh :  \(\frac{4}{3}+\frac{7}{6}+\frac{11}{10}+\frac{16}{15}+\frac{22}{21}+\frac{29}{28}+\frac{37}{36}+\frac{46}{45}\)

Answer 1

[giang ho dai ca]

=(1+1+1+1+1+1+1+1)+(1/3+1/6+1/10+1/15+1/21+1/28+1/36+1/45)

Đặt A = 1/3+1/6+1/10+1/15+1/21+1/28+1/36+1/45

Ta có:

A x 1/2= 1/6+1/12+1/20+1/30+1/42+1/56+1/72+1/90

1/6=1/2x3=1/2-1/3

1/12=1/3x4=1/3-1/4

……………………

1/90=1/9x10=1/9-1/10

A x 1/2=1/2-1/3+1/3-1/4+1/4-1/5+…+1/9-1/10

A x 1/2=1/2-1/10=4/10

A=4/10:1/2=4/5

Vậy 4/3+7/6+11/10+16/15+22/21+29/28+37/36+46/45=1+1+1+1+1+1+1+1+4/5=8+4/5=44/5

Answer 2

[Đinh Tuấn Việt]

\(\frac{4}{3}+\frac{7}{6}+\frac{11}{10}+...+\frac{46}{45}\)

\(=1+\frac{1}{3}+1+\frac{1}{6}+1+\frac{1}{10}+...+1+\frac{1}{45}\)

\(=\left(1+1+1+...+1\right)+\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{45}\right)\)(8 chữ số 1)

\(=8+\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{45}\right)\)

  Đặt A = \(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{45}\)

  => \(\frac{1}{2}\)A = \(\frac{1}{2}\times\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{45}\right)\)

              = \(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{90}\)

              \(=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{9.10}\)

              \(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{9}-\frac{1}{10}\)

              \(=\frac{1}{2}-\frac{1}{10}=\frac{2}{5}\)

Vậy A = \(\frac{2}{5}:\frac{1}{2}=\frac{4}{5}\)

    Do đó biểu thức trên là 8 + \(\frac{4}{5}\) = \(\frac{44}{5}\)

                                         Đáp số: \(\frac{44}{5}\)