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\(\frac{2}{n}+\frac{2}{n+1}=\frac{2\left(n+1\right)}{n\left(n+1\right)}+\frac{2n}{n\left(n+1\right)}\)\(=\frac{2\left(n+1\right)+2n}{n\left(n+1\right)}=\frac{2n+2+2n}{n\left(n+1\right)}=\frac{4n+2}{n\left(n+1\right)}\)
\(\frac{1}{n\left(n+1\right)}+\frac{-2}{n+1}=\frac{1}{n\left(n+1\right)}+\frac{-2n}{n\left(n+1\right)}\)\(=\frac{1+\left(-2n\right)}{n\left(n+1\right)}=\frac{1-2n}{n\left(n+1\right)}\)
a) \(A=\frac{135}{135.136-1}\) và \(B=\frac{136}{136.137-1}\)
\(A=\frac{1}{136-1}=\frac{1}{135}\) \(B=\frac{1}{137-1}=\frac{1}{136}\)
Vì \(\frac{1}{136}\)< \(\frac{1}{135}\)nên A > B.
a, A = \(\frac{136-1}{\left(136-1\right)136-1}\) = \(\frac{136-1}{136^2-136-1}\) B=\(\frac{136}{136\left(136+1\right)-1}\)=\(\frac{136}{136^2+136-1}\)
x=136, A-B =\(\frac{x-1}{x^2-x-1}\)-\(\frac{x}{x^2+x-1}\) =\(\frac{x^3+x^2-x-x^2-x+1-x^3+x^2+x}{\left(x^2-1\right)^2-x^2}\)=\(\frac{x^2-x+2}{\left(x^2-1\right)^2-x^2}\)<0
=> A<B
b,A = \(\frac{456-333}{456}\)= 1-333/456 B=\(\frac{789-333}{789}\)= 1-333/789
=> A>B
c, 3/14<3/13<3/12<3/11<3/10 <2/5
M = 3/10+3/11+3/12+3/13+3/14 < 2/5 x5 = 2= N
a) \(\frac{5}{9}=\frac{20}{36};\frac{1}{4}=\frac{9}{36}\)
\(\frac{20}{36}>\frac{9}{36}\Rightarrow\frac{5}{9}>\frac{1}{4}\)
\(\frac{72}{73}=\frac{4248}{4307};\frac{58}{59}=\frac{4234}{4307}\)
\(\frac{4248}{4307}>\frac{4234}{4307}\Rightarrow\frac{72}{73}>\frac{58}{59}\)
\(\frac{n}{n+3}=\frac{n+1}{n-1}=\frac{n+1}{3-2}=\frac{n+1}{n+2}\)
\(\Rightarrow\frac{n}{n+3}=\frac{n+1}{n+2}\)