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TA CÓ :
\(B=\frac{2010+2011+2012}{2011+2012+2013}\)
\(B=\frac{2010}{2011+2012+2013}+\frac{2011}{2011+2012+2013}+\frac{2012}{2011+2012+2013}\)
VÌ : \(\frac{2010}{2011}>\frac{2010}{2011+2012+2013}\)
\(\frac{2011}{2012}>\frac{2011}{2011+2012+2013}\)
\(\frac{2012}{2013}>\frac{2012}{2011+2012+2013}\)
=> A > B
VẬY , A > B
Mình tự hỏi. sao banh biết rồi còn đăng lên làm gì??????????
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)
\(\Rightarrow B>A\)
ta có: \(\frac{2011}{2012}>\frac{2011}{2012+2013};\frac{2012}{2013}>\frac{2012}{2013+2012}.\)
\(\Rightarrow A>\frac{2011}{2012+2013}+\frac{2012}{2013+2012}=\frac{2011+2012}{2012+2013}=B\)
....
Ta có \(\frac{2011}{2012}>\frac{2011}{2012+2013}\)
\(\frac{2012}{2013}>\frac{2012}{2012+2013}\)
CỘNG VẾ THEO VẾ,TA CÓ:
\(\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011}{2012+2013}+\frac{2012}{2012+2013}\)
\(\Rightarrow\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011+2012}{2012+2013}\)
\(\Rightarrow A>B\)
Vậy A>B
\(\frac{2010}{2011}\)> \(\frac{2010}{2011+2012+2013}\)
\(\frac{2011}{2012}\)> \(\frac{2011}{2011+2012+2013}\)
\(\frac{2012}{2013}\)> \(\frac{2012}{2011+2012+2013}\)
=> \(\frac{2010}{2011}\)+ \(\frac{2011}{2012}\)+ \(\frac{2012}{2013}\)> \(\frac{2010+2011+2012}{2011+2012+2013}\)
=> P > Q
Ta có
\(\frac{A^{2011}}{A^{2012}}=\frac{A^{2012}}{A^{2103}}=\frac{A}{A^2}\)
=> \(\frac{A^{2011}}{A^{2012}}+\frac{A^{2012}}{A^{2013}}=\frac{2A}{A^2}\)
\(\frac{A^{2011+2012}}{A^{2012+2013}}=\frac{A^{4023}}{A^{4025}}=\frac{1}{A^2}\)
=> \(\frac{A^{2011+2012}}{A^{2012+2013}}< \frac{A^{2011}}{A^{2012}}+\frac{A^{2012}}{A^{2013}}\)
Ta có :
\(Q=\frac{2010+2011+2012}{2011+2012+2013}=\frac{2010}{2011+2012+2013}+\frac{2011}{2011+2012+2013}+\frac{2012}{2011+2012+2013}\)
Vì :
\(\frac{2010}{2011}>\frac{2010}{2011+2012+2013}\)
\(\frac{2011}{2012}>\frac{2011}{2011+2012+2013}\)
\(\frac{2012}{2013}>\frac{2012}{2011+2012+2013}\)
Nên \(\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2013}>\frac{2010}{2011+2012+2013}+\frac{2011}{2011+2012+2013}+\frac{2012}{2011+2012+2013}\)
\(\Rightarrow\)\(\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2013}>\frac{2010+2011+2012}{2011+2012+2013}\)
\(\Rightarrow\)\(P>Q\)
Vậy \(P>Q\)
Chúc bạn học tốt ~
P = \(\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2013}\)
Q = \(\frac{2010+2011+2012}{2011+2012+2013}\) = \(\frac{2010}{2011+2012+2013}+\frac{2011}{2011+2012+2013}+\frac{2012}{2011+2012+2013}\)
Vì: \(\frac{2010}{2011}>\frac{2010}{2011+2012+2013}\)
\(\frac{2011}{2012}>\frac{2011}{2011+2012+2013}\)
\(\frac{2012}{2013}>\frac{2012}{2011+2012+2013}\)
=> \(\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2013}>\frac{2010}{2011+2012+2013}+\frac{2011}{2011+2012+2013}+\frac{2012}{2011+2012+2013}\)
P > Q
Ta có : Q=2010/2011+2012+2013 + 2011/2011+2012+2013 +2012/2011+2012+2013
Đó là bước đầu còn phần sau bạn tự so sánh từng phân số của P và Q nhé, k cho mik!
Ta có :
A=\(\frac{2011+2012}{2012+2013}=\frac{2011}{2012+2013}+\frac{2012}{2012+2013}\left(1\right)\)
B=\(\frac{2011}{2012}+\frac{2012}{2013}\left(2\right)\)
Từ (1) và (2) suy ra A<B
Đầu tiên:
Ta có:
B=\(\frac{2011}{2012+2013}\)+ \(\frac{2012}{2012+2013}\) = \(\frac{2011+2012}{2012+2013}\)
Vì:
\(\frac{2011}{2012+2013}\)< \(\frac{2011}{2012}\); \(\frac{2012}{2012+2013}\)< \(\frac{2012}{2013}\)
\(\Rightarrow\)\(\frac{2011+2012}{2012+2013}\)< \(\frac{2011}{2012}\)+ \(\frac{2012}{2013}\)
Mà \(\frac{2011+2012}{2012+2013}\)= B; \(\frac{2011}{2012}\)+ \(\frac{2012}{2013}\)
Vậy B>A