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\(a)\) Ta có :
\(\frac{1}{100}A=\frac{100^{2009}+1}{100^{2009}+100}=\frac{100^{2009}+100}{100^{2009}+100}-\frac{99}{100^{2009}+100}=1-\frac{99}{100^{2009}+100}\)
\(\frac{1}{100}B=\frac{100^{2010}+1}{100^{2010}+100}=\frac{100^{2010}+100}{100^{2010}+100}-\frac{99}{100^{2010}+100}=1-\frac{99}{100^{2010}+100}\)
Vì \(\frac{99}{100^{2009}+100}>\frac{99}{100^{2010}+100}\) nên \(1-\frac{99}{100^{2009}+100}< 1-\frac{99}{100^{2010}+100}\)
Do đó :
\(\frac{1}{100}A< \frac{1}{100}B\)\(\Rightarrow\)\(A< B\)
Vậy \(A< B\)
Chúc bạn học tốt ~
1.
\(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{99}}+\frac{1}{2^{100}}+\frac{1}{2^{100}}\)
\(=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}+\left(\frac{1}{2^{100}}+\frac{1}{2^{100}}\right)\)
\(=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}+\frac{1}{2^{99}}\)
cứ làm như vậy ta được :
\(=1+1=2\)
2. Ta có :
\(\frac{2008+2009}{2009+2010}=\frac{2008}{2009+2010}+\frac{2009}{2009+2010}\)
vì \(\frac{2008}{2009}>\frac{2008}{2009+2010}\); \(\frac{2009}{2010}>\frac{2009}{2009+2010}\)
\(\Rightarrow\frac{2008}{2009}+\frac{2009}{2010}>\frac{2008+2009}{2009+2010}\)
a) Ta có: \(\frac{1}{2^2}< \frac{1}{1.2}\) ; \(\frac{1}{3^2}< \frac{1}{2.3}\) ; \(\frac{1}{4^2}< \frac{1}{3.4}\) ; ... ; \(\frac{1}{2010^2}< \frac{1}{2009.2010}\)
=> \(Vt< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2009.2010}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2009}-\frac{1}{2010}\)
\(=1-\frac{1}{2010}< 1\)
\(2B=\frac{2+2^2+2^3+.....+2^{2010}}{1-2^{2010}}\)
\(B=2B-B=\left(2+2^2+2^3+.....+2^{2010}\right)-\left(2+2^2+2^3+.....+2^{2009}\right)\)
\(B=\frac{2^{2010}-1}{1-2^{2010}}\)
Ra được B rồi thì bạn tự so sánh với \(\frac{-1}{2}nghen\)
Ta có : \(A=1+2+2^2+.....+2^{2008}\)
\(=2^{2009}-1\)
Mà \(B=2^{2009}-1\)
=> \(A=B\)
Ta có: A = 1 + 2 + 22 + ... + 22008
=> 2A = 2 + 22 + 23 + ...+22009
=> 2A - A = (2 + 22 + 23 + ... + 22009) - (1 + 2 + 22 + ... + 22008)
=> A = 22009 - 1
Mà B = 22009 - 1
=> A = B
Vậy A = B
Ta có: A = 1 + 2 + 22+ ... +22008
=> 2A = 2 + 22+ ... +22009
=> 2A - A = 22009 - 1
=> A = 22009 - 1
Vậy A = B
A = 1 + 2 + 22 + ... + 22008
2A = 2 + 22 + 23 + ... + 22009
2A - A = ( 2 + 22 + 23 + ... + 22009) - ( 1 + 2 + 22 + ... + 22008)
A = 22009 - 1 = B
Vậy A = B
A = 1 + 2 + 22 + ... + 22008
2A = 2 + 22 + 23 + ... + 22009
2A - A = ( 2 + 22 + 23 +.... + 22009 ) - ( 1 + 2 + 22 + .... + 22008 )
A = 22009 - 1
Vậy A = B
Ta có : A = 1 + 2 + 22 + .....+ 22008
=> 2A = 2 + 22 + ....+ 22009
=> 2A - A = 22009 - 1
A = 22009 - 1
B = 22009 - 1
Vậy A = B