A=\(\frac{6}{4}+\frac{6}{28}+\frac{6}{70}+\frac{6}{130}+\frac{6}{208}\)

A=\(\frac{6}{1\cdot4}+\frac{6}{4\cdot7}+\frac{6}{7\cdot10}+\frac{6}{10\cdot13}+\frac{6}{13\cdot16}\)

A:2=\(\frac{3}{1\cdot4}+\frac{3}{4\cdot7}+\frac{3}{7\cdot10}+\frac{3}{10\cdot13}+\frac{3}{13\cdot16}\)

A:2=\(\frac{1}{1}-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+\frac{1}{10}-\frac{1}{13}+\frac{1}{13}-\)\(\frac{1}{16}\)

A:2=\(1-\frac{1}{16}\)

A:2=\(\frac{15}{16}\)

A=\(\frac{15}{8}\)

vậy ...

Kết quả là

A = 15/8

   Đs...

\(A=\frac{6}{4}+\frac{6}{28}+\frac{6}{70}+\frac{6}{130}+\frac{1}{208}\)

\(\Rightarrow A=\frac{6}{1\cdot4}+\frac{6}{4\cdot7}+\frac{6}{7\cdot10}+\frac{6}{10\cdot13}+\frac{6}{13\cdot16}\)

\(\Rightarrow\)\(\frac{3}{6}A=\frac{3}{6}\cdot\left(\frac{6}{1\cdot4}+\frac{6}{4\cdot7}+\frac{6}{7\cdot10}+\frac{6}{10\cdot13}+\frac{1}{13\cdot16}\right)\)

\(\Rightarrow\)\(\frac{3}{6}A=\frac{3}{1\cdot4}+\frac{3}{4\cdot7}+\frac{3}{7\cdot10}+\frac{3}{10\cdot13}+\frac{3}{13\cdot16}\)

\(\Rightarrow\)\(\frac{3}{6}A=1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+\frac{1}{10}-\frac{1}{13}+\frac{1}{13}-\frac{1}{16}\)

\(\Rightarrow\)\(\frac{3}{6}A=1-\frac{1}{16}\)

\(\Rightarrow\)\(\frac{3}{6}A=\frac{15}{16}\)

\(\Rightarrow A=\frac{15}{16}:\frac{3}{6}\Rightarrow A=\frac{15}{8}\)

Bạn ơi, có chắc là 6/280 ở cuối không.

Trả lời nhanh để mink giải câu này cho

lộn \(\frac{6}{208}\) chứ ko phải 6/ 280

\(\frac{6}{1.4}+\frac{6}{4.7}+\frac{6}{7.10}+.....+\frac{6}{13.16}\)

Ta có công thức \(\frac{a}{b.c}=\frac{a}{c-b}.\left(\frac{1}{b}-\frac{1}{c}\right)\)

Dựa vào công thức ta có :

\(\frac{6}{1.4}=\frac{6}{3}.\left(1-\frac{1}{4}\right)\)

\(\frac{6}{4.7}=\frac{6}{3}.\left(\frac{1}{4}-\frac{1}{7}\right)\)

.....................................

\(\frac{6}{13.16}=\frac{6}{3}.\left(\frac{1}{13}-\frac{1}{16}\right)\)

\(\Rightarrow\)\(\frac{6}{4}+\frac{6}{28}+\frac{6}{70}+......+\frac{6}{208}=\frac{6}{3}.\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+.....+\frac{1}{13}-\frac{1}{16}\right)\)

\(\Rightarrow\)\(\frac{6}{4}+\frac{6}{28}+\frac{6}{70}+.....+\frac{6}{208}=\frac{6}{3}.\left(1-\frac{1}{16}\right)=\frac{6}{3}.\left(\frac{15}{16}\right)=\frac{15}{8}\)

Mink chắc chắn, ai thấy đúng thì ủng hộ mink nha !!!

Bài 1: 

\(=\dfrac{1}{\left(x+1\right)\left(x+4\right)}+\dfrac{1}{\left(x+4\right)\left(x+7\right)}+\dfrac{1}{\left(x+7\right)\left(x+10\right)}+\dfrac{1}{\left(x+10\right)\left(x+13\right)}+\dfrac{1}{\left(x+13\right)\left(x+16\right)}\)

\(=\dfrac{1}{3}\left(\dfrac{3}{\left(x+1\right)\left(x+4\right)}+\dfrac{3}{\left(x+4\right)\left(x+7\right)}+\dfrac{3}{\left(x+7\right)\left(x+10\right)}+\dfrac{3}{\left(x+10\right)\left(x+13\right)}+\dfrac{3}{\left(x+13\right)\cdot\left(x+16\right)}\right)\)

\(=\dfrac{1}{3}\left(\dfrac{1}{x+1}-\dfrac{1}{x+4}+\dfrac{1}{x+4}-\dfrac{1}{x+7}+\dfrac{1}{x+7}-\dfrac{1}{x+10}+\dfrac{1}{x+10}-\dfrac{1}{x+13}+\dfrac{1}{x+13}-\dfrac{1}{x+16}\right)\)

\(=\dfrac{1}{3}\left(\dfrac{1}{x+1}-\dfrac{1}{x+16}\right)\)

\(=\dfrac{1}{3}\cdot\dfrac{x+16-x-1}{\left(x+1\right)\left(x+16\right)}=\dfrac{5}{\left(x+1\right)\left(x+16\right)}\)

Bài 2: 

\(\Leftrightarrow a^2-2a+1+b^2+4b+4+4c^2-4c+1=0\)

\(\Leftrightarrow\left(a-1\right)^2+\left(b+4\right)^2+\left(2c-1\right)^2=0\)

Dấu '=' xảy ra khi a=1; b=-4; c=1/2

c)  \(A=\frac{6}{4}+\frac{6}{28}+\frac{6}{70}+\frac{6}{130}+\frac{6}{208}\) 

\(=\frac{6}{1.4}+\frac{6}{4.7}+\frac{6}{7.10}+\frac{6}{10.13}+\frac{6}{13.16}\) 

\(=2\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+\frac{1}{10}-\frac{1}{13}+\frac{1}{13}-\frac{1}{16}\right)\)

\(=2\left(1-\frac{1}{16}\right)\) 

\(=2.\frac{15}{16}\) 

\(=\frac{15}{8}\) 

Vậy A=\(\frac{15}{8}\)

a) \(\frac{3^2}{1.4}+\frac{3^2}{4.7}+\frac{3^2}{7.10}+...+\frac{3^2}{97.100}\)

\(=3\left(\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{97.100}\right)\)

\(=3\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{97}-\frac{1}{100}\right)\)

\(=3\left(1-\frac{1}{100}\right)\)

\(=3.\frac{99}{100}=\frac{297}{100}\)

\(\frac{1}{10}+\frac{1}{40}+\frac{1}{88}+\frac{1}{154}+\frac{1}{238}+\frac{1}{340}\)

\(=\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+\frac{1}{11.14}+\frac{1}{14.17}+\frac{1}{17.20}\)

\(=\frac{1}{2}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}+...+\frac{1}{17}-\frac{1}{20}\)

\(=\frac{1}{2}-\frac{1}{20}\)

\(=\frac{10}{20}-\frac{1}{20}\)

\(=\frac{9}{20}\)

A = \(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{56}\)

\(=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{7.8}\)

  \(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+....+\frac{1}{7}-\frac{1}{8}\)

  \(=\frac{1}{2}-\frac{1}{8}=\frac{3}{8}\)

B = \(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{11.13}\)

  \(=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{11}-\frac{1}{13}\)

  \(=1-\frac{1}{13}=\frac{12}{13}\)

\(A=\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{56}\)

\(=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{7.8}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{7}-\frac{1}{8}\)

\(=\frac{1}{2}-\frac{1}{8}=\frac{3}{8}\)

\(B=\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{11.13}\)

\(=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{11}-\frac{1}{13}\)

\(=1-\frac{1}{13}=\frac{12}{13}\)

A=\(\frac{1}{6}\)+\(\frac{1}{12}\)+......+\(\frac{1}{56}\)

A=\(\frac{1}{2.3}\)+\(\frac{1}{3.4}\)+...+\(\frac{1}{7.8}\)

A=\(\frac{1}{2}\)-\(\frac{1}{3}\)+\(\frac{1}{3}\)-\(\frac{1}{4}\)+....+\(\frac{1}{7}\)-\(\frac{1}{8}\)

A=\(\frac{1}{2}\)-\(\frac{1}{8}\)=\(\frac{3}{8}\)

B=\(\frac{2}{1.3}\)+\(\frac{2}{3.5}\)+...+\(\frac{2}{11.13}\)

B=1-\(\frac{1}{3}\)+\(\frac{1}{3}\)-\(\frac{1}{5}\)+....+\(\frac{1}{11}\)-\(\frac{1}{13}\)

B=1-\(\frac{1}{13}\)=\(\frac{12}{13}\)

C=\(\frac{3}{4}\)+\(\frac{3}{28}\)+....+\(\frac{3}{304}\)

C=\(\frac{3}{1.4}\)+\(\frac{3}{4.7}\)+..+\(\frac{3}{16.19}\)

Rồi bạn cũng tách ra như câu A và câu B

1/2A=1/2(6/4+6/28+6/70+6/130+6/208)

       = 3/4+3/28+3/70+3/130+3/208

       = 1-1/4+1/4-1/7+..................-1/16

       =1-1/16

       =15/16 => A=15/8

=(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

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

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

\(=\left(1+1+1+...+1\right)+\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{45}\right)\)(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: \(\frac{44}{5}\)

hic hic, Việt , mình đồng cảm với bạn

mình đánh chữ chậm như rùa

Đặt P = ... ( biểu thức đề bài ) 

Nhận xét: Với \(k\inℕ^∗\) ta có: 

\(\frac{k+2}{k!+\left(k+1\right)!+\left(k+2\right)!}=\frac{k+2}{k!+\left(k+1\right).k!+\left(k+2\right).k!}=\frac{k+2}{2.k!\left(k+2\right)}=\frac{1}{2.k!}\)

\(\Rightarrow\)\(P=\frac{1}{2.1!}+\frac{1}{2.2!}+...+\frac{1}{2.6!}=\frac{1}{2}\left(1+\frac{1}{2}+...+\frac{1}{720}\right)=...\)