Cho A =\(\dfrac{100^{2007}+1}{100^{2008}+1}\) và B = \(\dfrac{100^{2006}+1}{100^{2007+1}}\).Hãy so sánh A và B
Cho A =\(\frac{100^{2007}+1}{100^{2008}+1}\)và cho B=\(\frac{100^{2006}+1}{100^{2007}+1}\).Hãy so sánh A và B
\(A=\frac{100^{2007}+1}{100^{2008}+1}\Rightarrow100.A=\frac{100^{2008}+100}{100^{2008}+1}=\frac{100^{2008}+1+99}{100^{2008}+1}=1+\frac{99}{100^{2008}+1}\)
\(B=\frac{100^{2006}+1}{100^{2007}+1}\Rightarrow100.B=\frac{100^{2007}+100}{100^{2007}+1}=\frac{100^{2007}+1+99}{100^{2007}+1}=1+\frac{99}{100^{2007}+1}\)
Vì \(\frac{99}{100^{2007}+1}>\frac{99}{100^{2008}+1};1=1\Rightarrow1+\frac{99}{100^{2007}+1}>1+\frac{99}{100^{2008}+1}\)hay \(A>B\)
Vậy \(A>B\)
So sánh A và B biết:\(A=\frac{100^{2007}+1}{100^{2008}+1};B=\frac{100^{2006}+1}{100^{2007}+1}\)
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Cho A= 1002007+1/1002008+1 ; B= 1002006+1/1002007+1
1. So sánh A và B
A= 20082007 +1/ 20082008 + 1
B= 20082007 + 1/ 20082008 +1
2. So sánh M và N
M= 100100 + 1/ 10099 +1
N= 100101 +1/ 100100+1
3. Cm:
B= 5^2008 +5^2007 +5^2006 chia hết cho 31.
C= 8^8 +2^20 chia hết cho 17.
D= 313^5 . 299- 313^6 . 36 chia hết cho 7
a) c/m
1/1001+1/1002+1/1003+...+1/2000 > 13/21
b) cho A=1002007+1/1002008+1
B=1002006+1/1002007+1
so sanh A va B
gjup ho nha, nhanh nhanh mk dng rat can
Thanks
a) chứng minh rằng 1/22 + 1/32 + 1/42 + ...... + 1/20082 < 1
b) cho A= 1002007 + 1/ 1002008 +1; B= 1002006 + 1/ 1002007 +1. hãy so sánh A và B?
c) S= 1/31+1/32+...+1/60. chứng minh: 3/5 < S < 4/5
So sánh A và B biết : \(A=\dfrac{10^{2006}+1}{10^{2007}+1},B=\dfrac{10^{2007}+1}{10^{2008}+1}\)
So sánh A và B, biết \(A=\dfrac{10^{2006}+1}{10^{2007}+1};B=\dfrac{10^{2007}+1}{10^{2008}+1}\)
\(10A=\dfrac{10^{2007}+10}{10^{2007}+1}=\dfrac{10^{2007}+1+9}{10^{2007}+1}=1+\dfrac{9}{10^{2007}+1}\left(1\right)\)\(10B=\dfrac{10^{2008}+10}{10^{2008}+1}=\dfrac{10^{2008}+1+9}{10^{2008}+1}=1+\dfrac{9}{10^{2008}+1}\left(2\right)\)Từ (1) và ( 2 ) suy ra A>B
Cách 2 :
Ta CM BĐT sau :
\(\dfrac{a}{b}< \dfrac{a+m}{b+m}\left(a< b;a;b;m>0\right)\)
Ta có :
\(a< b\\ \Rightarrow am< bm\\ \Rightarrow ab+am< bm+ab\\ \Rightarrow a\left(b+m\right)< b\left(a+m\right)\\ \Rightarrow\dfrac{a}{b}< \dfrac{a+m}{b+m}\)
\(\Rightarrow A=\dfrac{10^{2007}+1}{10^{2008}+1}< \dfrac{10^{2007}+1+9}{10^{2008}+1+9}\\ =\dfrac{10\left(10^{2006}+1\right)}{10\left(10^{2007}+1\right)}=\dfrac{10^{2006}+1}{10^{2007}+1}=B\\ \Rightarrow A< B\)
So Sánh
a,A= \(\frac{2008^{2008}+1}{2008^{2009}+1}\)và B=\(\frac{2008^{2007}+1}{2008^{2008}+1}\)
b, M=\(\frac{100^{100}+1}{100^{99}+1}\)và N= \(\frac{100^{101}+1}{100^{100}+1}\)
a) Áp dụng \(\frac{a}{b}< 1\Leftrightarrow\frac{a}{b}< \frac{a+m}{b+m}\) (a;b;m \(\in\) N*)
Ta có:
\(A=\frac{2008^{2008}+1}{2008^{2009}+1}< \frac{2008^{2008}+1+2007}{2009^{2009}+1+2007}\)
\(A< \frac{2008^{2008}+2008}{2008^{2009}+2008}\)
\(A< \frac{2008.\left(2008^{2007}+1\right)}{2008.\left(2008^{2008}+1\right)}=\frac{2008^{2007}+1}{2008^{2008}+1}=B\)
=> A < B
b) Áp dụng \(\frac{a}{b}>1\Leftrightarrow\frac{a}{b}>\frac{a+m}{b+m}\) (a;b;m \(\in\) N*)
Ta có:
\(N=\frac{100^{101}+1}{100^{100}+1}>\frac{100^{101}+1+99}{100^{100}+1+99}\)
\(N>\frac{100^{101}+100}{100^{100}+100}\)
\(N>\frac{100.\left(100^{100}+1\right)}{100.\left(100^{99}+1\right)}=\frac{100^{100}+1}{100^{99}+1}=M\)
=> M > N