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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 )
PT
0
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
\(A< \frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\frac{1}{9.11}+\frac{1}{11.13}+\frac{1}{13.15}.\)
\(2A< \frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+\frac{2}{9.11}+\frac{2}{11.13}+\frac{2}{13.15}\)
\(2A< \frac{3-1}{1.3}+\frac{5-3}{3.5}+\frac{7-5}{5.7}+\frac{9-7}{7.9}+\frac{11-9}{9.11}+\frac{13-11}{11.13}+\frac{15-13}{13.15}\)
\(2A< 1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{13}-\frac{1}{15}=1-\frac{1}{15}< 1\)
\(\Rightarrow2A< 1\Rightarrow A< \frac{1}{2}\)
BC
2
27 tháng 4 2018
\(Đ\text{ặt }S=\frac{1}{4}+\frac{1}{16}+\frac{1}{36}+....+\frac{1}{10000}\)
\(S=\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+...+\frac{1}{100^2}\)
\(S=\frac{1}{2^2}\cdot\left(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\right)\)
Ta có :
\(\frac{1}{2^2}< \frac{1}{1\cdot2};\text{ }\frac{1}{3^2}< \frac{1}{2\cdot3};\text{ }...;\text{ }\frac{1}{50^2}< \frac{1}{49\cdot50}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+...+\frac{1}{49\cdot50}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)
\(=1-\frac{1}{50}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}< 1\Rightarrow1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}< 1+1=2\)
\(\Rightarrow\frac{1}{2^2}\left(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\right)< \frac{1}{2^2}\cdot2\)
\(\Rightarrow S< \frac{1}{2}\) (ĐPCM)
SK
30 tháng 4 2018
Đặt \(A=\frac{1}{4}+\frac{1}{16}+\frac{1}{36}+....+\frac{1}{10000}\)
\(\Rightarrow A=\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{100^2}\)
\(\Rightarrow4A=1+\frac{1}{2^2}+\frac{1}{3^2}+....+\frac{1}{50^2}\)
\(\Rightarrow4A< 1+\frac{1}{1.2}+\frac{1}{2.3}+....+\frac{1}{49.50}\)
\(\Rightarrow4A=1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)
\(\Rightarrow4A< 2-\frac{1}{50}< 2\)
\(\Rightarrow4A< 2\Rightarrow A< \frac{2}{4}=\frac{1}{2}\)
=>a<1/2
NQ
4
25 tháng 4 2019
Bạn tham khảo nhé
A=14 +116 +136 +164 +1100 +1144 +1196 =122 +142 +162 +182 +1102 +1122 +1142
2A=222 +242 +262 +282 +2102 +2122 +2142
2A<12 +22.4 +24.6 +26.8 +28.10 +210.12 +212.14
2A<12 +12 −14 +14 −16 +16 −18 +18 −110 +110 −112 +112 −114
2A<12 +12 −114
2A<1−114
2A<1314
A<1328 <1428 =12 ( đpcm )
Vậy A<12
Chúc bạn học tốt ~
`= 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