(1-1/4). (1-1/9). 1- (1/16).....(1-1/144)
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AQ
0
H
1
11 tháng 1 2018
Sửa đề : \(S=\left(1-\frac{1}{4}\right).\left(1-\frac{1}{9}\right).\left(1-\frac{1}{16}\right).....\left(1-\frac{1}{144}\right)\)
\(\Rightarrow S=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}.....\frac{143}{144}\)
\(\Rightarrow S=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}.....\frac{11.13}{12.12}\)
\(\Rightarrow S=\frac{1.2.3.....11}{2.3.4.....12}.\frac{3.4.5.....13}{2.3.4.....12}\)
\(\Rightarrow S=\frac{1}{12}.\frac{13}{2}\)
\(\Rightarrow S=\frac{13}{24}\)
LH
3
KN
9 tháng 8 2020
\(P=\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+...+\frac{1}{121}+\frac{1}{144}\)
\(\Rightarrow P=\frac{1}{4}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+...+\frac{1}{11^2}+\frac{1}{12^2}\)
Ta có : \(P< \frac{1}{4}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{10.11}+\frac{1}{11.12}\)
\(\Rightarrow P< \frac{1}{4}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{10}-\frac{1}{11}+\frac{1}{11}-\frac{1}{12}\)
\(\Rightarrow P< \frac{1}{4}+\frac{1}{2}-\frac{1}{12}\)
\(\Rightarrow P< \frac{2}{3}\left(đpcm\right)\)
NH
9 tháng 8 2020
\(P=\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+...+\frac{1}{121}+\frac{1}{144}\)
\(P=\frac{1}{4}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+...+\frac{1}{11^2}+\frac{1}{12^2}\)
Có : \(P< \frac{1}{4}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{10.11}+\frac{1}{11.12}\)
\(\Rightarrow P< \frac{1}{4}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{10}-\frac{1}{11}+\frac{1}{11}-\frac{1}{12}\)
\(\Rightarrow P< \frac{1}{4}=\frac{1}{2}-\frac{1}{12}\)
\(\Rightarrow P< \frac{2}{3}\)( đpcm )
HA
28 tháng 10 2017
a, 1+2+4+8+16+32+...+256+512+1024
= 20+21+22+23+24+25+...+28+29+210
= 1 + 2 + 22+3+4+5+...+8+9+10
= 3+254
DP
17 tháng 7 2017
\(\left(1-\frac{1}{4}\right)\cdot\left(1-\frac{1}{9}\right)\cdot\left(1-\frac{1}{16}\right)\cdot...\cdot\left(1-\frac{1}{444}\right)\cdot\left(1-\frac{1}{169}\right)\)
\(=\frac{3}{4}\cdot\frac{8}{9}\cdot\frac{15}{16}\cdot...\cdot\frac{143}{144}\cdot\frac{168}{169}\)
\(=\frac{1.3}{2.2}\cdot\frac{2.4}{3.3}\cdot\frac{3.5}{4.4}\cdot....\cdot\frac{11.13}{12.12}\cdot\frac{12.14}{13.13}\)
\(=\frac{1\cdot2\cdot3\cdot...\cdot11\cdot12}{2\cdot3\cdot4\cdot...\cdot12\cdot13}\cdot\frac{3\cdot4\cdot5\cdot....\cdot13\cdot14}{2\cdot3\cdot4\cdot....\cdot12\cdot13}\)
\(=\frac{1}{13}\cdot\frac{14}{2}=\frac{7}{13}\)
DD
1
5 tháng 12 2021
\(=1^2+2^2+3^2+4^2+...+12^2+13^2\\ =\dfrac{13\times\left(13+1\right)\times\left(13\times2+1\right)}{6}=\dfrac{13\times14\times27}{6}=819\)
DH
1
NN
Nguyễn Ngọc Anh Minh
CTVHS
VIP
25 tháng 2 2017
1/144 PHẢI ĐỔI THÀNH 1/100 MỚI ĐÚNG HƠN. BẠN XEM LẠI ĐỀ BÀI XEM
\(A< \frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\frac{1}{9.11}+\frac{1}{11.13}+\frac{1}{13.15}.\)
\(2A< \frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+\frac{2}{9.11}+\frac{2}{11.13}+\frac{2}{13.15}\)
\(2A< \frac{3-1}{1.3}+\frac{5-3}{3.5}+\frac{7-5}{5.7}+\frac{9-7}{7.9}+\frac{11-9}{9.11}+\frac{13-11}{11.13}+\frac{15-13}{13.15}\)
\(2A< 1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{13}-\frac{1}{15}=1-\frac{1}{15}< 1\)
\(\Rightarrow2A< 1\Rightarrow A< \frac{1}{2}\)
`= 3/4 . 8/9 . 15/16. ... . 143/144`
`= (1.2.3...12)/(2.3.4....12) . (3.4.....12.13)/(2.3....11.12)`
`= 1/12 . 13/2 = 13/24`
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)*...*(1-1/12)(1+1/12)
=1/2*2/3*...*11/12*3/2*4/3*...*13/12
=1/12*13/2=13/24