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\(5A=\frac{5}{1.6}+\frac{5}{6.11}+\frac{5}{11.16}+...+\frac{5}{\left(5n+1\right).\left(5n+6\right)}\)

\(5A=1-\frac{1}{6}+\frac{1}{6}-\frac{1}{11}+\frac{1}{11}-\frac{1}{16}+...+\frac{1}{5n+1}-\frac{1}{5n+6}\)

\(5A=1-\frac{1}{5n+6}=\frac{5n+6-1}{5n+6}=\frac{5n+5}{5n+6}\)=> \(A=\frac{n+1}{5n+6}\)

C = 1/1 . 6 + 1/6 . 11 + 1/11 . 16 + ...+ 1/( 5n + 1 ) . ( 5n + 6 ) 

C = 1/5 . ( 5/1 . 6 + 5/6 . 11 + 5/11 . 16 + ...+ 5/( 5n + 1 ) . ( 5n + 6 )  ) 

C = 1/5 . ( 1 - 1/6 + 1/6 - 1/11 + 1/11 - 1/16 + ...+ 1/5n + 1 - 1/5n + 6  ) 

C = 1/5 . ( 1 - 1/5n + 6 ) 

C = 1/5 . 1 - 1/5 . 1/5n + 6

C = 1/5 - 1/ 5 . ( 5n + 6 ) 

mình trả lời bài 1 thôi nhé :

Gọi biểu thức trên là A.

Theo bài ra ta có:A=1/1.6+1/6.11+1/11.16+...+1/(5n+1)+1/(5n+6)

                           =1/5(1-1/6+1/6-1/11+1/11-1/16+...+1/5n+1-1/5n+6)

                           =1/5(1-1/5n+6)

                           =1/5( 5n+6/5n+6-1/5n+6)

                           =1/5(5n+6-1/5n+6)

                           =1/5.5n+5/5n+6

                           =n+1/5n+6

                           =ĐIỀU PHẢI CHỨNG MINH

x- 20/11.13 - 20/13.15 - 20/13.15 - 20/15.17 -...- 20/53.55=3/11

x-10.(2/11.13+2/13.15+2/15.17+...+2/53.55=3/11

x-10.(1/11-1/13+1/13-1/15+1/15-1/17+...+1/53-1/55)=3/11

x-10.(1/11-1/55)=3/11

x-10.4/55=3/11

x-8/11=3/11

x = 3/11+8/11

x=11/11=1

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ban Optimus Prime sai dau bai rui

Ta có:

\(\frac{1}{1.6}+\frac{1}{6.11}+...+\frac{1}{\left(5n+1\right)\left(5n+6\right)}=\frac{1}{5}\left(\frac{1}{1}-\frac{1}{6}+\frac{1}{6}-\frac{1}{11}+...+\frac{1}{5n+1}-\frac{1}{5n+6}\right)\)

\(=\frac{1}{5}\left(\frac{1}{1}-\frac{1}{5n+6}\right)=\frac{1}{5}\left(\frac{5n+6}{5n+6}-\frac{1}{5n+6}\right)=\frac{1}{5}.\frac{5n+5}{5n+6}=\frac{1}{5}.\frac{5\left(n+1\right)}{5n+6}=\frac{5\left(n+1\right)}{5\left(5n+6\right)}=\frac{n+1}{5n+6}\)(ĐPCM)

bạn Phạm Thiết Tường ơi ch mình hỏi sao lại nhân \(\frac{1}{5}\)với \(\frac{1}{1}-\frac{1}{5n+6}\)vậy

@tranlethanhtruc: Bn ấy lm sai.

 CM:  \(\dfrac{1}{1.6}\)+ \(\dfrac{1}{11.16}\)+...+ \(\dfrac{1}{\left(5n+1\right)\left(5n+6\right)}\) = \(\dfrac{n+1}{5n+6}\)

A = \(\dfrac{1}{5}\)(\(\dfrac{5}{1.6}\) + \(\dfrac{5}{6.11}\)+...+ \(\dfrac{5}{\left(5n+1\right).\left(5n+6\right)}\)) 

A = \(\dfrac{1}{5}\).( \(\dfrac{1}{1}\) - \(\dfrac{1}{6}\)+ \(\dfrac{1}{6}\) - \(\dfrac{1}{11}\)+...+ \(\dfrac{1}{5n+1}\) - \(\dfrac{1}{5n+6}\))

A = \(\dfrac{1}{5}\) .( \(\dfrac{1}{1}\) - \(\dfrac{1}{5n+6}\))

A = \(\dfrac{1}{5}\). \(\dfrac{5n+6-1}{5n+6}\)

A = \(\dfrac{1}{5}\). \(\dfrac{5n+5}{5n+6}\)

A = \(\dfrac{1}{5}\) . \(\dfrac{5.\left(n+1\right)}{5n+6}\)

A = \(\dfrac{n+1}{5n+6}\)

⇒\(\dfrac{1}{1.6}\) + \(\dfrac{1}{6.11}\)+ \(\dfrac{1}{11.16}\)+...+ \(\dfrac{1}{\left(5n+1\right)\left(5n+6\right)}\) = \(\dfrac{n+1}{5n+1}\) (đpcm)

\(A=\dfrac{1}{1.6}+\dfrac{1}{6.11}+\dfrac{1}{11.16}+...+\dfrac{1}{\left(5n+1\right)\left(5n+6\right)}\)

\(A=\dfrac{1}{5}\left[1-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{16}+...+\dfrac{1}{5n+1}-\dfrac{1}{5n+6}\right]\)

\(A=\dfrac{1}{5}\left(1-\dfrac{1}{5n+6}\right)\)

\(A=\dfrac{1}{5}\left(\dfrac{5n+6-1}{5n+6}\right)=\dfrac{1}{5}\left(\dfrac{5n+5}{5n+6}\right)=\dfrac{1}{5}.5\left(\dfrac{n+1}{5n+6}\right)=\dfrac{n+1}{5n+6}\)

\(\Rightarrow dpcm\)