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Ta có:
\(M=\frac{101^{102}+1}{101^{103}+1}\)
\(101M=\frac{101^{103}+1+100}{101^{103}+1}=1+\frac{100}{101^{103}+1}\)
Ta lại có:
\(N=\frac{101^{103}+1}{101^{104}+1}\)
\(101N=\frac{101^{104}+1+100}{101^{104}+1}=1+\frac{100}{101^{104}+1}\)
Vì \(\frac{100}{101^{104}+1}< \frac{100}{101^{103}+1}\Rightarrow101N< 101M\Rightarrow N< M\)
Ta có : \(101M=\frac{101\left(101^{102}+1\right)}{101^{103}+1}=\frac{101^{103}+100+1}{101^{103}+1}=1+\frac{100}{101^{103}+1};\)
\(101N=\frac{101\left(101^{103}+1\right)}{101^{104}+1}=\frac{101^{104}+1+100}{101^{104}+1}=1\frac{100}{101^{104}+1}\)
Vì \(\frac{100}{101^{103}+1}>\frac{100}{101^{104}+1}\Rightarrow1+\frac{100}{101^{103}+1}>1+\frac{100}{101^{104}+1}\Rightarrow101M>101N\)
=> M > N
So sánh M và N biết rằng :
\(M=\frac{101^{102}+1}{101^{103}+1}\)
\(N=\frac{101^{103}+1}{101^{104}+1}\)
ta có bổ đề sau .với\(\frac{a}{b}>0\Rightarrow\frac{a}{b}< \frac{a+c}{b+c}\)
\(\Rightarrow N=\frac{101^{103}+1}{101^{104}+1}< \frac{101^{103}+1+100}{101^{104}+1+100}\)
mà \(\frac{101^{103}+1+100}{101^{104}+1+100}=\frac{101^{103}+101}{101^{104}+101}\)
\(=\frac{101\left(101^{102+1}\right)}{101\left(101^{103}+1\right)}=\frac{101^{102}+1}{101^{103}+1}=M\)
vậy \(M>N\)
Ta có: \(N=\frac{101^{103}+1}{101^{104}+1}< \frac{101^{103}+1+100}{101^{104}+1+100}\)
Mà: \(\frac{101^{103}+1+100}{101^{104}+1+100}=\frac{101^{103}+101}{101^{104}+101}=\frac{101\left(101^{102}+1\right)}{101\left(101^{103}+1\right)}=\frac{101^{102}+1}{101^{103}+1}=M\)
Ta có: \(N< \frac{101^{103}+1+100}{101^{104}+1+100};\frac{101^{103}+1+100}{101^{104}+1+100}=M\)
=> N<M
=>
N = 101^103 + 1 / 101^104 + 1 < 101^103 + 1 + 100 / 101^104 + 1 + 100
= 101^103 + 101 / 101^104 + 101
= 101(101^102 + 1) / 101(101^103 + 1)
= 101^102 + 1 / 101^103 + 1 = M
=> N < M
101M=101(101^102+1)/101^103+1
=101^103+1+100/101^103+1
=1+100/101^103+1
101N=101(101^103+1)/101^104+1
=101^104+1+100/101^104+1=1+100/101^104+1
THẤY;100/101^104+1<100/101^103+1
nên;M>N
Ta có :
\(N=\frac{101^{103}+1}{101^{104}+1}< 1=\frac{101^{103}+1+100}{101^{104}+1+100}=\frac{101^{103}+101}{101^{104}+101}=\frac{101\left(101^{102}+1\right)}{101\left(101^{103}+1\right)}=\frac{101^{102}+1}{101^{103}+1}=M\)
Vậy\(N< M\)
Ta có: M =\(\frac{101^{102}+1}{101^{103}+1}=\frac{101^{103}+101}{101^{104}+101}=\frac{101^{103}+1+100}{101^{104}+1+100}\)
Mà : N = \(\frac{101^{103}+1}{101^{104}+1}\)< M = \(\frac{101^{103}+1+100}{101^{104}+1+100}\)
\(\Rightarrow N< M\)
\(.....^{2014}\)thì lun lun \(>....^{2013}\)rồi
Đặt 10M và 10N rồi làm