NP
3
K
Khách
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NV
Nguyễn Việt Lâm
Giáo viên
7 tháng 5 2021
Ta có:
\(\dfrac{n^2-1}{n^2}=1-\dfrac{1}{n^2}>1-\dfrac{1}{\left(n-1\right)n}\)
Áp dụng:
\(C=\dfrac{2^2-1}{2^2}+\dfrac{3^2-1}{3^2}+\dfrac{4^2-1}{4^2}+...+\dfrac{100^2-1}{100^2}\)
\(C>1-\dfrac{1}{1.2}+1-\dfrac{1}{2.3}+1-\dfrac{1}{3.4}+...+1-\dfrac{1}{99.100}\)
\(C>99-\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{99.100}\right)\)
\(C>99-\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{99}-\dfrac{1}{100}\right)\)
\(C>99-\left(1-\dfrac{1}{100}\right)\)
\(C>98+\dfrac{1}{100}>98\) (đpcm)
NV
0
NY
1
WH
4 tháng 5 2018
Trả lời
Ta có:
\(C=\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+...+\frac{9999}{10000}\)
\(\Rightarrow C=\left(1-\frac{1}{4}\right)+\left(1-\frac{1}{9}\right)+\left(1-\frac{1}{16}\right)+...+\left(1-\frac{1}{10000}\right)\)
\(\Rightarrow C=\left(1+1+1+...+1\right)-\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{10000}\right)\)(99 chữ số 1)
\(\Rightarrow C=99-\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{10000}\right)\)
Ta lại có:
\(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{10000}=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)
Đặt D\(=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)
\(\Rightarrow D< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{99\cdot100}\)
\(\Rightarrow D< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(\Rightarrow D< 1-\frac{1}{100}\)
\(\Rightarrow D< \frac{99}{100}< 1\)
\(\Rightarrow C>99-1\)
\(\Rightarrow C>98\)
Vậy C>98 (đpcm)
MA
2
BT
12 tháng 5 2017
\(C=\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+...+\frac{9999}{10000}\)
Ta có: \(\frac{3}{4}=1-\frac{1}{4};\frac{8}{9}=1-\frac{1}{9};\frac{15}{16}=1-\frac{1}{16};...;\frac{9999}{10000}=1-\frac{1}{10000}\)
=> \(C=1-\frac{1}{4}+1-\frac{1}{9}+1-\frac{1}{16}+...+1-\frac{1}{10000}\)
=> \(C=\left(1+1+1+...+1\right)-\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\right)\)(99 chữ số 1)
=> \(C=99-\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\right)\)
Ta lại có: \(\frac{1}{2^2}< \frac{1}{1.2}=1-\frac{1}{2}\); \(\frac{1}{3^2}< \frac{1}{2.3}=\frac{1}{2}-\frac{1}{3}\); \(\frac{1}{4^2}>\frac{1}{3.4}=\frac{1}{3}-\frac{1}{4}\); ...;\(\frac{1}{100^2}< \frac{1}{99.100}=\frac{1}{99}-\frac{1}{100}\)
=> \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< 1-\frac{1}{100}=\frac{99}{100}< 1\)
=> C > 99-1
=> C > 98
12 tháng 5 2017
Ta có :C=
\(=\frac{2^2-1}{2^2}+\frac{3^2-1}{3^2}+\frac{4^2-1}{4^2}+...+\frac{100^2-1}{100^2}\)
\(=\frac{2^2}{2^2}+\frac{3^2}{3^2}+...+\frac{100^2}{100^2}-\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}\right)\)
\(=99-\)\(\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}\right)\)
Mà \(\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}\right)\)l<\(\frac{100}{101}\)(tự tính)
Suy ra C> 98(đpcm)
AT
27 tháng 4 2017
Ta có: \(A=\left\{\dfrac{3}{4}+\dfrac{8}{9}+\dfrac{15}{16}+...+\dfrac{9999}{10000}\right\}\Rightarrow99\)số
\(A=\left(1-\dfrac{1}{4}\right)+\left(1-\dfrac{1}{9}\right)+...+\left(1-\dfrac{1}{100000}\right)\)
\(A=\left\{1+1+1+...+1\right\}\Rightarrow99\)số \(-\dfrac{1}{4}+\dfrac{1}{9}+\dfrac{1}{16}+...+\dfrac{1}{100000}=99-\left(\dfrac{1}{4}+\dfrac{1}{9}+\dfrac{1}{16}+...+\dfrac{1}{10000}\right)\)
Ta có: \(4=2^2>1.2\Rightarrow\dfrac{1}{4}< \dfrac{1}{1.2}\Leftrightarrow\dfrac{1}{4}< \dfrac{1}{1}-\dfrac{1}{2}\)
Tương tự: \(\dfrac{1}{9}< \dfrac{1}{2}-\dfrac{1}{3};\dfrac{1}{16}< \dfrac{1}{3}-\dfrac{1}{4};...;\dfrac{1}{10000}< \dfrac{1}{99}-\dfrac{1}{100}\)
Cộng theo vế ta được: \(\dfrac{1}{4}+\dfrac{1}{9}+\dfrac{1}{16}+\dfrac{1}{10000}< \dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{99}-\dfrac{1}{100}=1-\dfrac{1}{100}< 1\)
\(\Rightarrow A=99-\left(\dfrac{1}{4}+\dfrac{1}{6}+\dfrac{1}{16}+...+\dfrac{1}{10000}\right)>99-1=98\)
Vậy \(A>98\)
BT
19 tháng 4 2018
Ta có: \(\frac{3}{4}=1-\frac{1}{4}=1-\frac{1}{2^2}\); \(\frac{8}{9}=1-\frac{1}{9}=1-\frac{1}{3^2}\)
\(\frac{15}{16}=1-\frac{1}{16}=1-\frac{1}{4^2}\); ...; \(\frac{9999}{10000}=1-\frac{1}{10000}=1-\frac{1}{100^2}\)
=> \(C=\left(1+1+...+1\right)-\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\right)\)
=> \(C=99-\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\right)=99-B\)
\(B=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)
=> \(B< \frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}=1-\frac{1}{100}< 1\)
=> A > 99-1 = 98
=> B > 98
5 tháng 3 2018
Đặt \(A=\frac{3}{4}+\frac{8}{9}+..........+\frac{9999}{10000}\)
\(=\left(1-\frac{1}{4}\right)+\left(1-\frac{1}{9}\right)+..........+\left(1-\frac{1}{10000}\right)\)
\(=1-\frac{1}{2^2}+1-\frac{1}{3^2}+...........+1-\frac{1}{100^2}\)
\(=99-\left(\frac{1}{2^2}+\frac{1}{3^2}+......+\frac{1}{100^2}\right)\)\(>99-\left(\frac{1}{1.2}+\frac{1}{2.3}+..........+\frac{1}{99.100}\right)\)
\(=99-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.........+\frac{1}{99}-\frac{1}{100}\right)\)
\(=99-\left(1-\frac{1}{100}\right)=99-1+\frac{1}{100}=98+\frac{1}{100}>98\)
N
1
3 tháng 8 2017
C = ( 1 - 1/4 ) + ( 1 - 1/9 ) + ( 1 - 1/16 ) + .. .+ ( 1 - 1/10000 )
C = 1 + 1 + ... + 1 - ( 1/4 + 1/9 + 1/16 + ... + 1/10000 )
C = 1 + 1 + 1 +... + 1 - ( 1/22 + 1/32 + .. + 1/1002 )
C = 99 - ( 1/22 + 1/32 + ... + 1/1002 )
Mà 1/22 + 1/32 + ... + 1/1002 < 1/1.2 + 1/2.3 + 1/3.4 + ... + 1/99.100 = 1 - 1/2 + 1/2 - 1/3 + .. + 1/99 - 1/100 = 1 - 1/100 < 1 =>
C > 99 - 1 => C > 98
S
3
10 tháng 5 2017
\(C=\frac{4-1}{4}+\frac{9-1}{9}+....+\frac{10000-1}{10000}.\)
\(C=1-\frac{1}{4}+1-\frac{1}{9}+.....+1-\frac{1}{10000}.\)
\(C=\left(1+1+....+1\right)-\left(\frac{1}{4}+\frac{1}{9}+...+\frac{1}{10000}\right)\)
\(C=99-\left(\frac{1}{4}+\frac{1}{9}+...+\frac{1}{10000}\right)\)
ta có :\(\frac{1}{4}< 1,\frac{1}{9}< 1,......,\frac{1}{10000}< 1\)
\(\Rightarrow\frac{1}{4}+\frac{1}{9}+...+\frac{1}{10000}< 1\)
\(C=99-\left(\frac{1}{4}+\frac{1}{9}+...+\frac{1}{10000}\right)>98\)
vậy C>98
S
10 tháng 5 2017
\(C=\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+...+\frac{9999}{10000}\)
\(=\left(1-\frac{1}{4}\right)+\left(1-\frac{1}{9}\right)+\left(1-\frac{1}{16}\right)+...+\left(1-\frac{1}{10000}\right)\)
\(=\left(1+1+1+...+1\right)-\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{10000}\right)\)
\(=99-\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\right)\)
Đặt D = \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)
Ta có: \(\frac{1}{2^2}< \frac{1}{1.2}\)
\(\frac{1}{3^2}< \frac{1}{2.3}\)
\(\frac{1}{4^2}< \frac{1}{3.4}\)
.............
\(\frac{1}{100^2}< \frac{1}{99.100}\)
\(\Rightarrow D>\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+..+\frac{1}{99.100}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}=1-\frac{1}{100}< 1\)
\(\Rightarrow C=99-\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\right)>98\)(đpcm)
đề đúng rồi
\(C=\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+...+\frac{9999}{10000}\)
\(C=\left(1-\frac{1}{4}\right)+\left(1-\frac{1}{9}\right)+\left(1-\frac{1}{16}\right)+...+\left(1-\frac{1}{10000}\right)\)
\(C=\left(1+1+1+...+1\right)-\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{10000}\right)\)
\(C=99-\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{10000}\right)\)
đặt \(A=\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{10000}\)
\(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)
\(A< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(=1-\frac{1}{100}=\frac{99}{100}< 1\)
\(\Rightarrow A< 1\)
Vì \(A< 1\)nên \(B=99-\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{10000}\right)>99-1=98\)
= 3/22 + 8/32 + 15/42 + ... + 9999/1002
= 1.3/2.2 + 2.4/3.3 + 3.5/4.4 + .... + 99.101/100.100
\(=\frac{1.3.2.4.3.5.4.6...99.101}{2^2.3^2....100^2}=\frac{1.2.3^2.4^2...99^2.100.101}{2^2.3^2....100^2}=\frac{1.2.101}{2^2.100}=\frac{101}{200}\)