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Ta có:
\(2006A=\dfrac{2006^{2007}+2016}{2006^{2007}+1}=1+\dfrac{2005}{2006^{2007}+1}\)
\(2006B=\dfrac{2006^{2006}+2006}{2006^{2006}+1}=1+\dfrac{2005}{2006^{2006}+1}\)
Do \(\dfrac{2005}{2006^{2006}+1}>\dfrac{2005}{2006^{2007}+1}\Rightarrow1+\dfrac{2005}{2006^{2006}+1}>1+\dfrac{2005}{2006^{2007}+1}\)
\(\Rightarrow2006A< 2006B\Rightarrow A< B\)
Mình sẽ giải cách ngắn hơn cách bạn đạt nha:
Nếu:
\(\dfrac{a}{b}< 1\Rightarrow\dfrac{a+m}{b+m}< 1\left(m\in N\right)\)
\(A=\dfrac{2006^{2006}+1}{2006^{2007}+1}< 1\)
\(A< \dfrac{2006^{2006}+1+2005}{2006^{2007}+1+2005}\Rightarrow A< \dfrac{2006^{2006}+2006}{2006^{2007}+2006}\Rightarrow A< \dfrac{2006\left(2006^{2005}+1\right)}{2006\left(2006^{2006}+1\right)}\Rightarrow A< \dfrac{2006^{2005}+1}{2006^{2006}+1}=B\)\(A< B\)
Nếu:
\(\dfrac{a}{b}< 1\Rightarrow\dfrac{a+m}{b+m}< 1\left(m\in N\right)\)
\(A=\dfrac{2005^{2005}+1}{2005^{2006}+1}< 1\)
\(A< \dfrac{2005^{2005}+1+2004}{2005^{2006}+1+2004}\Rightarrow A< \dfrac{2005^{2005}+2005}{2005^{2006}+2005}\Rightarrow A< \dfrac{2005\left(2005^{2004}+1\right)}{2005\left(2005^{2005}+1\right)}\Rightarrow A< \dfrac{2005^{2004}+1}{2005^{2005}+1}=B\)
\(A< B\)
Ta có : A = \(\dfrac{2005^{2005}+1}{2005^{2006}+1}\)
\(2005\)A = \(\dfrac{\left(2005^{2005}+1\right).2005}{2005^{2006}+1}\)
\(2005\)\(A\)= \(\dfrac{2005^{2006}+2005}{2005^{2006}+1}\)
\(2005\)\(A\)= \(\dfrac{2005^{2006}+1+2004}{2005^{2006}+1}\)
\(2005A=\dfrac{2005^{2006}+1}{2005^{2006}+1}+\dfrac{2004}{2005^{2006}+1}\)
\(2005A=1+\dfrac{2004}{2005^{2006}+1}\)
Tương tự như vậy với \(B\) ta đc
\(2005B=1+\dfrac{2004}{2005^{2005}+1}\)
Vì \(2005^{2006}+1>2005^{2005}+1\)
\(=>\) \(1+\dfrac{2004}{2005^{2006}+1}\)\(< \)\(1+\dfrac{2004}{2005^{2005}+1}\)
\(=>\)\(2005A< 2005B\)
\(=>\)\(A< B\)
Vậy \(A< B\)
\(C=\dfrac{2006\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2007}\right)}{\left(1+\dfrac{2005}{2}\right)+\left(1+\dfrac{2004}{3}\right)+...+\left(1+\dfrac{1}{2006}\right)+1}\)
\(=\dfrac{2006\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2007}\right)}{\dfrac{2007}{2}+\dfrac{2007}{3}+...+\dfrac{2007}{2007}}=\dfrac{2006}{2007}\)
Lời giải:
Ta có:
\(N=\dfrac{-7}{10^{2005}}+\dfrac{-15}{10^{2006}}=\dfrac{-7}{10^{2005}}+\dfrac{-7}{10^{2006}}+\dfrac{-8}{10^{2006}}\)
\(M=\dfrac{-15}{10^{2005}}+\dfrac{-7}{10^{2006}}=\dfrac{-7}{10^{2005}}+\dfrac{-8}{10^{2005}}+\dfrac{-7}{10^{2006}}\)
Xét \(N\) và \(M\) có \(\dfrac{-7}{10^{2005}}+\dfrac{-7}{10^{2006}}\) chung.
Mà \(\dfrac{-8}{10^{2005}}>\dfrac{-8}{10^{2006}}\) nên \(N>M\).
Áp dụng Bất đẳng thức :
\(\dfrac{a}{b}< 1\Leftrightarrow\dfrac{a}{b}< \dfrac{a+m}{b+m}\)
Ta có :
\(\dfrac{2006^{2006}+1}{2006^{2007}+1}< \dfrac{2006^{2006}+1+2005}{2006^{2007}+1+2005}=\dfrac{2006^{2006}+2006}{2006^{2007}+2006}=\dfrac{2006\left(2006^{2005}+1\right)}{2006\left(2006^{2006}+1\right)}=\dfrac{2006^{2005}+1}{2006^{2006}+1}\)
\(\Leftrightarrow\dfrac{2006^{2006}+1}{2006^{2007}+1}< \dfrac{2006^{2005}+1}{2006^{2006}+1}\)
Ta có:
\(2005A=\dfrac{2005^{2006}+2005}{2005^{2006}+1}=1+\dfrac{2004}{2005^{2006}+1}\)
\(2005B=\dfrac{2005^{2005}+2005}{2005^{2005}+1}=1+\dfrac{2004}{2005^{2005}+1}\)
Vì \(\dfrac{2004}{2005^{2006}+1}< \dfrac{2004}{2005^{2005}+1}\Rightarrow1+\dfrac{2004}{2005^{2006}+1}< 1+\dfrac{2004}{2005^{2005}+1}\)
\(\Rightarrow2005A< 2005B\Rightarrow A< B\)
Vậy A < B
Chị sử dụng cách làm lớp 7 ở câu 3 nha em
em cũng tự quy đồng và suy ra cách làm của cô giáo dạy em nha
chữ cj xấu thì mong em thông cảm
