Đặt \(\frac{12}{1.4.7}+\frac{12}{4.7.10}+...+\frac{12}{54.57.60}=A\)

\(\frac{A}{2}=\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{54.57.60}\)

\(\frac{A}{2}=\frac{7-1}{1.4.7}+\frac{10-4}{4.7.10}+...+\frac{60-54}{54.57.60}\)

\(\frac{A}{2}=\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...+\frac{1}{54.57}-\frac{1}{57.60}=\frac{1}{1.4}-\frac{1}{57.60}\)

\(A=\frac{1}{2}-\frac{1}{30.57}< \frac{1}{2}\)

Gọi biểu thức là A, ta có:

A = \(\frac{12}{1.4.7}+\frac{12}{4.7.10}+\frac{12}{7.10.13}+...+\frac{12}{54.57.60}=2\left(\frac{6}{1.4.7}+\frac{6}{4.7.10}+\frac{6}{7.10.13}+...+\frac{6}{54.57.60}\right)\)

A = \(2\left(\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+\frac{1}{7.10}-\frac{1}{10.13}+...+\frac{1}{54.57}-\frac{1}{57.60}\right)\)

A = \(2\left(\frac{1}{1.4}-\frac{1}{57.60}\right)=2\left(\frac{427}{1710}\right)=\frac{427}{855}< \frac{427}{854}=\frac{1}{2}\)

Vậy A < \(\frac{1}{2}\)(điều cần chứng minh)

Câu hỏi của thục hà - Toán lớp 6 - Học toán với OnlineMath

Em tham khảo nhé!

Đề sai hả

\(P=\frac{12}{1.4.7}+\frac{12}{4.7.10}+...+\frac{12}{54.57.60}\)

\(\Rightarrow\frac{1}{2}P=\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{54.57.60}\)

\(\Rightarrow\frac{1}{2}P=\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...+\frac{1}{54.57}-\frac{1}{57.60}\)

\(\Rightarrow\frac{1}{2}P=\frac{1}{1.4}-\frac{1}{57.60}< \frac{1}{4}\)

\(\Rightarrow P< \frac{1}{4}.2=\frac{1}{2}\)

\(P=\frac{12}{1.4.7}+\frac{12}{4.7.10}+...+\frac{12}{54.57.60}\)

\(=2\left(\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{54.57.60}\right)\)

\(=2\left(\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...+\frac{1}{54.57}-\frac{1}{57.60}\right)\)

\(=2\left(\frac{1}{4}-\frac{1}{3420}\right)\)

\(=2\left(\frac{855-1}{3420}\right)\)

\(=2.\frac{427}{1710}=\frac{427}{855}\)

Mà \(\frac{1}{2}=\frac{427}{854}\)

=> \(\frac{427}{855}< \frac{427}{854}\)=> P < \(\frac{1}{2}\)

\(P=\frac{12}{1.4.7}+\frac{12}{4.7.10}+\frac{12}{7.10.13}+...+\frac{12}{54.57.60}\)

\(P=4.\left(\frac{3}{1.4.7}+\frac{3}{4.7.10}+\frac{3}{7.10.13}+...+\frac{3}{54.57.60}\right)\)

\(P=4\left(\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+\frac{1}{7.10}+...+\frac{1}{54.57}-\frac{1}{57.60}\right)\)

\(P=4.\left(\frac{1}{4}-\frac{1}{3420}\right)\)

\(P=4.\frac{427}{1710}\)

\(P=\frac{854}{855}\)

nhớ nhiều nhé

duyệt đi mà nhanh lên sốt ruột quá

=2.(6/1.4.7 + 6/4.7.10 + 6/7.10.13 + ... + 6/54.57.60)

=2.(1/1.4-1/4.7+1/4.7-1/7.10+1/7.10-1/10.13+...+1/54.57-1/57.60)

=2(1.4-1/57.60)

TỰ TÍNH

P = 2*[ 6/(1*4*7) + 6/(4*7*10) + ... + 6/(54*57*60) ]

   = 2*[ 1/(1*4) - 1/(4*7) + 1/(4*7) - 1/(7*10) + ... + 1/(54*57) -1/(57*60) ]

   = 2*[ 1/(1*4) - 1/(57*60) ]

   = 2* (427/1710)

   = 427/855 <1/2

S = 1+ 1/2^2 + 1/3^2 +... + 1/100^2

1/2^2 < 1/(1*2)

1/3^2 < 1/(2*3)

...

1/100^2 < 1/(99*100)

==> 1/2^2 +1/3^2 +.., +1/100^2 < 1/(1*2) + 1/(2*3) + ... + 1/(99*100) = 1 -1/2 +1/2 - 1/3 +1/3 -1/4 +... - 1/100

                                                                                                   =1 - 1/100 <1

==> 1/2^2 + 1/3^2 +... + 1/100^2  < 1

==> 1 + 1/2^2 + 1/3^2 +... +1/100^2 <2