1.tính
B= 12/ 1.4.7 + 12/ 4.7.10 + 12/ 7.10.13 +....+ 12/54.57.60
2.chứng minh
a) A= 1+ 1/2^2 + 1/3^2 +...+1/100^2 < 2
b) B= 1/5 + 1/13 +1/25 +...+1/n^2+(n+1)^2< 1/2
Chứng minh rằng:
\(\frac{12}{1.4.7}+\frac{12}{4.7.10}+\frac{12}{7.10.13}+...+\frac{12}{54.57.60}< \frac{1}{2}\)
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)
chứng minh rằng : P = \(\frac{12}{1.4.7}+\frac{12}{4.7.10}+\frac{12}{7.10.13}+...+\frac{12}{54+57+60}< \frac{1}{2}\)
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 < \(\frac{1}{2}\)
Cho \(P=\frac{12}{1.4.7}+\frac{12}{4.7.10}+\frac{12}{7.10.13}+.....+\frac{12}{55.58.61}SosánhVớiP\frac{1}{2}\)
CMR:
A= 1/3 + 1/3 + 1/3 +...+ 1/3 < 1/2
B= 1 + 1/2 + 1/3 + ...+ 1/100 <2
C= 12/ 1.4.7 + 12/ 4.7.10 + ...+ 12/54.57.60 < 1/2
D= 1.2/ 1.4.7 + 1.2/ 4.7.10 +...+ 1.2/ 54.57.60 < 1/2
Lưu ý: C và D là hai đề bài khác nhau nhưng có khi mình chép lộn dấu nên mình mới làm thành hai câu C và D nếu ra đáp án có lẽ mình chép đúng. Tks!!'
Chứng minh rằng:
a) P = \(\frac{12}{1.4.7}\)+\(\frac{12}{4.7.10}\)+\(\frac{12}{7.10.13}\)+...+\(\frac{12}{54.57.60}\)<\(\frac{1}{2}\)
b) S = 1+\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{100^2}\)<2
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
BT6 : Cho A = 124 . ( 1/1.1985 + 1/2.1986 + 1/3.1987 +......+1/16.200 )
B = 1/1.17 + 1/20.18 +....+1/1984.2000
So sánh A và B
BT7: Cho P = 12/1.4.7 + 12/4.7.10 + 12/7.10.13 +....+12/54.57.60
So sánh P và 1/2
Giúp mik nha các bạn
BT 7 :
\(P=\frac{12}{1.4.7}+\frac{12}{4.7.10}+\frac{12}{7.10.13}+...+\frac{12}{54.57.60}\)
\(P=\frac{12}{6}\left(\frac{6}{1.4.7}+\frac{6}{4.7.10}+\frac{6}{7.10.13}+...+\frac{6}{54.57.60}\right)\)
\(P=2\left(\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+\frac{1}{7.10}-\frac{1}{10.13}+...+\frac{1}{54.57}-\frac{1}{57.60}\right)\)
\(P=2\left(\frac{1}{1.4}-\frac{1}{57.60}\right)\)
\(P=2\left(\frac{1}{4}-\frac{1}{3420}\right)\)
\(P=\frac{1}{2}-\frac{1}{1710}< \frac{1}{2}\)
Vậy \(P< \frac{1}{2}\)
Chúc bạn học tốt ~
CHỨNG MINH RẰNG:\(\frac{12}{1.4.7}+\frac{12}{4.7.10}+...+\frac{12}{54.57.60}< \frac{1}{2}\)
Đặt \(\frac{12}{1.4.7}+\frac{12}{4.7.10}+...+\frac{12}{54.57.60}=A\)
\(\frac{A}{2}=\frac{6}{1.4.7}+\frac{6}{4.7.10}+...+\frac{6}{54.57.60}\)
\(\frac{A}{2}=\frac{7-1}{1.4.7}+\frac{10-4}{4.7.10}+...+\frac{60-54}{54.57.60}\)
\(\frac{A}{2}=\frac{1}{1.4}-\frac{1}{4.7}+\frac{1}{4.7}-\frac{1}{7.10}+...+\frac{1}{54.57}-\frac{1}{57.60}=\frac{1}{1.4}-\frac{1}{57.60}\)
\(A=\frac{1}{2}-\frac{1}{30.57}< \frac{1}{2}\)
chứng minh rằng :
12 / 1.4.7 +12 / 4.7.10 +12 / 7.10.12 +...+12 / 54.57.60 <1/2
nhanh đúng like
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
Chứng Minh
\(\frac{12}{1.4.7}+\frac{12}{4.7.10}+\frac{12}{7.10.12}+.......+\frac{12}{54.57.60}<\frac{1}{2}\)