So sánh 2 phân số :
\(A=\frac{2012^{2012}+1}{2012^{2013}+1}\) và \(B=\frac{2012^{2011}+1}{2012^{2012}+1}\)
So sánh A và B biết : \(A=\frac{2012^{2012}+1}{2012^{2013}+1};B=\frac{2012^{2011}+1}{2012^{2012}+1}\)
\(10A=\frac{2012^{2013}+10}{2012^{2013}+1}=\frac{2012^{2013}+1+9}{2012^{2013}+1}=1+\frac{9}{2012^{2013}+1}\)
\(10B=\frac{2012^{2012}+10}{2012^{2012}+1}=\frac{2012^{2012}+1+9}{2012^{2012}+1}=1+\frac{9}{2012^{2012}+1}\)
Vì \(\frac{9}{2012^{2013}+1}
ta co A=\(\frac{2012^{2012}+1}{2012^{2013}+1}< \frac{2012^{2012}+1+2011}{2012^{2013}+1+2011}\)=\(\frac{2012^{2012}+2012}{2012^{2013}+2012}=\frac{2012\left(2012^{2011}+1\right)}{2012\left(2012^{2012}+1\right)}\)
=>A<B
So sánh 2 số sau: M=\(\frac{2013^{2012}+2012}{2013^{2011}+1}\)và \(N=\frac{2013^{2011}+2012}{2013^{2010}+1}\)
Ta có :
\(\frac{1}{2013}M=\frac{2013^{2012}+2012}{2013^{2012}+2013}=\frac{2013^{2012}+2013}{2013^{2012}+2013}-\frac{1}{2013^{2012}+2013}=1-\frac{1}{2013^{2012}+2013}\)
Lại có :
\(\frac{1}{2013}N=\frac{2013^{2011}+2012}{2013^{2011}+2013}=\frac{2013^{2011}+2013}{2013^{2011}+2013}-\frac{1}{2013^{2011}+2013}=1-\frac{1}{2013^{2011}+2013}\)
Vì \(\frac{1}{2013^{2012}+2013}< \frac{1}{2013^{2011}+2013}\) nên \(M=1-\frac{1}{2013^{2012}}>N=1-\frac{1}{2013^{2011}+2013}\)
Vậy \(M>N\)
Chúc bạn học tốt ~
So sánh: A=\(\frac{^{2012^{2012}}+1}{2012^{2013}+1}\)và B=\(\frac{2012^{2011}+1}{2012^{2012}+1}\)
A=\(\frac{2012^{2012}+1}{2012^{2013}+1}\)
\(\Rightarrow\)A<\(\frac{2012^{2012}+1+2011}{2012^{2013}+1+2011}\)
<\(\frac{2012^{2012}+2012}{2012^{2013}+2012}\)
<\(\frac{2012\left(2012^{2011}+1\right)}{2012\left(2012^{2012}+1\right)}\)
<\(\frac{2012^{2011}+1}{2012^{2012}+1}\)
<B
Vậy A<B
so sánh phân số
A=2012^2012+1 trên 2012^2013+1
B=2012^2011+1 trên 2012^2013+1
so sánh các phân số sau
\(\frac{2012\cdot2013}{2012\cdot2013+1}\)và\(\frac{2013}{2012}\)
\(\frac{2012}{2013}\)và \(\frac{2011}{2012}\)
Ta có \(\frac{2012.2013}{2012.2013+1}\)và \(\frac{2013}{2012}\)
Vì \(\frac{2012.2013}{2012.2013+1}< 1< \frac{2013}{2012}\)
nên \(\frac{2012.2013}{2012.2013+1}< \frac{2013}{2012}\)
\(\frac{2012}{2013}\)và \(\frac{2011}{2012}\)
phàn bù của \(\frac{2012}{2013}\)là \(\frac{1}{2013}\)
phàn bù của \(\frac{2011}{2012}\)là \(\frac{1}{2012}\)
Vì \(\frac{1}{2012}>\frac{1}{2013}\Rightarrow\frac{2012}{2013}>\frac{2011}{2012}\)
Ta có : \(\frac{2012\cdot2013}{2012\cdot2013+1}< 1\)
\(\frac{2013}{2012}>1\)
\(\Rightarrow\frac{2012\cdot2013}{2012\cdot2013+1}< \frac{2013}{2012}\)
Có : \(\frac{2012}{2013}=1-\frac{2012}{2013}=\frac{2013}{2013}-\frac{2012}{2013}=\frac{1}{2013}\)
\(\frac{2011}{2012}=1-\frac{2011}{2012}=\frac{2012}{2012}-\frac{2011}{2012}=\frac{1}{2012}\)
Vì \(2013< 2012\)nên \(\frac{1}{2013}< \frac{1}{2012}\)hay \(\frac{2012}{2013}< \frac{2011}{2012}\)
So sánh các phân số sau :
\(\frac{2012\times2013}{2012\times2013+1}\)và \(\frac{2013}{2012}\)
Ta có :
\(\frac{2012\times2013}{2012\times2013+1}\)
\(=\frac{1}{1}=1\)( cùng rút gọn 2012 x 2013 )
Vì \(1< \frac{2013}{2012}\)nên \(\frac{2012\times2013}{2012\times2013+1}< \frac{2013}{2013}\).
\(\frac{2012}{2013}\)và \(\frac{2011}{2012}\)
Ta có :
\(1-\frac{2012}{2013}=\frac{1}{2013};1-\frac{2011}{2012}=\frac{1}{2012}\)
Vì \(\frac{1}{2013}< \frac{1}{2012}\)nên \(\frac{2012}{2013}< \frac{2011}{2012}\)
~ Chúc bạn hok tốt ~
So sánh:
a) \(\frac{2^{10}+1}{2^{10}-1}\)và \(\frac{2^{10}-1}{2^{10}-3}\)
b) \(\frac{2011}{2012}+\frac{2012}{2013}\)và \(\frac{2011+2012}{2012+2013}\)
a) \(\frac{2^{10}+1}{2^{10}-1}\)và \(\frac{2^{10}-1}{2^{10}-3}\)
Ta có chính chất phân số trung gian là \(\frac{2^{10}+1}{2^{10}-3}\)
\(\frac{2^{10}+1}{2^{10}-1}>\frac{2^{10}+1}{2^{10}-3}\) ; \(\frac{2^{10}-1}{2^{10}-3}< \frac{2^{10}+1}{2^{10}-3}\)
Vì \(\frac{2^{10}+1}{2^{10}-1}>\frac{2^{10}+1}{2^{10}-3}>\frac{2^{10}-1}{2^{10}-3}\)
Nên \(\frac{2^{10}+1}{2^{10}-1}>\frac{2^{10}-1}{2^{10}-3}\)
b) \(A=\frac{2011}{2012}+\frac{2012}{2013}\)và \(B=\frac{2011+2012}{2012+2013}\)
Ta có : \(A=\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011}{2013}+\frac{2012}{2013}=\frac{2011+2012}{2013}>\frac{2011+2012}{2012+2013}=B\)
Vậy A > B
Có gì sai cho sorry
a,
\(\frac{2^{10}+1}{2^{10}-1}=1+\frac{2}{2^{10}-1}< 1+\frac{2}{2^{10}-3}=\frac{2^{10}-1}{2^{10}-3}\)
b,
\(\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011}{2012+2013}+\frac{2012}{2012+2013}=\frac{2011+2012}{2012+2013}\)
So sánh P và Q biết : P = 2010/2011 + 2011/2012 + 2012/2013 và Q = 2010+2011+2012/ 2011 +2012+2013
Chứng tỏ N < 1 với N = \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2009^2}+\frac{1}{2010^2}\)
Ta có: \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2010^2}
\(\frac{2012^{2010}+1}{2012^{2011}+1}và\frac{2012^{2011}+1}{2012^{2012}+1}\)so sánh 2 số
So sánh \(A=\frac{2011}{2012}+\frac{2012}{2013}\) và \(B=\frac{2011+2012}{2012+2013}\)
Ta có \(B=\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011}{2013}+\frac{2012}{2013}=\frac{2011+2012}{2013}\)
Lại có: \(\frac{2011+2012}{2013}>\frac{2011+2012}{2012+2013}\) ( ngoặc 2 dòng này lại nhé dòng này và dòng trên)
\(\Rightarrow B>A\)