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Ta có:
\(A-B=\dfrac{1999^{1999}+1}{1999^{1998}+1}-\dfrac{1999^{2000}+1}{1999^{1999}+1}\)
\(=\dfrac{\left(1999^{1999}+1\right)^2-\left(1999^{1998}+1\right)\left(1999^{2000}+1\right)}{\left(1999^{1998}+1\right)\left(1999^{1999}+1\right)}\)
\(=\dfrac{1999^{3998}+2\cdot1999^{1999}+1-\left(1999^{3998}+1999^{1998}+1999^{2000}+1\right)}{1999^{3997}+1999^{1998}+1999^{1999}+1}\)
\(=\dfrac{2\cdot1999^{1999}-1999^{1998}-1999^{2000}}{1999^{3997}+1999^{1998}+1999^{1999}+1}\)
Mà \(2\cdot1999^{1999}-1999^{1998}-1999^{2000}=-\left[\left(1999^{999}\right)^2-2\cdot1999^{999}\cdot1999^{1000}+\left(1999^{1000}\right)^2\right]\)
\(=-\left(1999^{999}-1999^{1000}\right)^2< 0\)
Mà mẫu số > 0
\(\Rightarrow A-B< 0\Leftrightarrow A< B\)
A=\(\dfrac{1999^{1999}+1999-1998}{1999^{1998}+1}\) B=\(\dfrac{1999^{2000}+1999-1998}{1999^{1999}+1}\)
A=1999-\(\dfrac{1998}{1999^{1998}+1}\) B=1999-\(\dfrac{1998}{1999^{1999}+1}\)
Vì 19991998+1<19991999+1 nên
\(\dfrac{1}{1999^{1998}+1}\)>\(\dfrac{1}{1999^{1999}+1}\) nên \(\dfrac{-1}{1999^{1998}+1}< \dfrac{-1}{1999^{1999}+1}\)
A=1999+\(\dfrac{-1}{1999^{1998}+1}< 1999+\dfrac{-1}{1999^{1999}+1}\)=B
A<B
a) +) Có \(A=\frac{13^{15}+1}{13^{16}+1}\)=> 13A = \(\frac{13\left(13^{15}+1\right)}{13^{16}+1}\)
= \(\frac{13^{16}+13}{13^{16}+1}=\frac{13^{16}+1+12}{13^{16}+1}=1+\frac{12}{13^{16}+1}\)(1)
+) Có \(B=\frac{13^{16}+1}{13^{17}+1}\)=> 13B =\(\frac{13\left(13^{16}+1\right)}{13^{17}+1}\)
=\(\frac{13^{17}+13}{13^{17}+1}=1+\frac{12}{13^{17}+1}\)(2)
+) Từ (1) và (2) => \(1+\frac{12}{13^{16}+1}>1+\frac{12}{13^{17}+1}\)
<=> 13A>13B <=> A> B
b) +) Có A=\(\frac{1999^{1999}+1}{1999^{1998}+1}\) => \(\frac{A}{1999}=\frac{1999^{1999}+1}{1999^{1999}+1999}=\frac{1999^{1999}+1999-1998}{1999^{1999}+1999}\)
=\(1-\frac{1998}{1999^{1999}+1999}\) (1)
+) Có B =\(\frac{1999^{2000}+1}{1999^{1999}+1}\)
=> \(\frac{B}{1999}=\frac{1999^{2000}+1}{1999^{2000}+1999}=1-\frac{1998}{1999^{2000}+1999}\)(2)
+) Từ (1) và (2) => \(1-\frac{1998}{1999^{1999}+1999}\)< \(1-\frac{1998}{1999^{2000}+1999}\)
<=> \(\frac{A}{1999}< \frac{B}{1999}\) <=> A< B
c: \(\dfrac{A}{10}=\dfrac{100^{100}+1}{100^{100}+10}=1-\dfrac{9}{100^{100}+10}\)
\(\dfrac{B}{10}=\dfrac{100^{69}+1}{100^{69}+10}=1-\dfrac{9}{100^{69}+10}\)
Ta có: 100^100+10>100^69+10
=>-9/(100^100+10)<-9/(100^69+10)
=>A/10<B/10
=>A<B
1/2000*1999 - 1/1999*1998 - 1/1998*1997 - ... - 1/2*1
= 1/1999 - 1/2000 - (1/1998 - 1/1999) - (1/1997 - 1/1998) - ... - (1 - 1/2)
= 1/1999 - 1/2000 - 1/1998 + 1/1999 - 1/1997 +1/1998 - .... - 1 + 1/2
= 1/1999 + 1/1999 - 1/2000 - 1/1998 + 1/1998 - 1/1997 +1/1997 - .... - 1/2 +1/2 - 1
= 1/1999 + 1/1999 - 1/2000 - 1
= 2/1999 - 1 - 1/2000
= -1997/1999 - 1/2000
= -2000 - 1997/1997*2000
=-3997/3994000
=
#)Giải :
Ta có :
\(A=\frac{1999^{1999}+1}{1999^{1998}+1}=\frac{1999^{1999}+1999-1998}{1999^{1998}+1}=1999-\frac{1998}{1999^{1998}+1}\)
\(B=\frac{1999^{2000}+1}{1999^{1999}+1}=\frac{1999^{2000}+1999-1998}{1999^{1999}+1}=1999-\frac{1998}{1999^{1999}+1}\)
Vì \(1999^{1998}+1< 1999^{1999}+1\)
\(\Rightarrow\frac{1}{1999^{1998}+1}>\frac{1}{1999^{1999}+1}\Rightarrow1999+\frac{-1}{1999^{1998}+1}< 1999+\frac{-1}{1999^{1999}+1}\Rightarrow A< B\)
a. Có: \(\frac{100^{101}+1}{100^{100}+1}>1\Rightarrow\frac{100^{101}+1}{100^{100}+1}>\frac{100^{101}+\left(1+99\right)}{100^{100}+\left(1+99\right)}\)
\(\Rightarrow B>\frac{100^{101}+100}{100^{100}+100}\\ \Rightarrow B>\frac{100\left(100^{100}+1\right)}{100\left(100^{99}+1\right)}\\ \Rightarrow B>\frac{100^{100}+1}{100^{99}+1}=A\\ \Leftrightarrow A< B\)
Vậy A < B
b. Có: \(\frac{13^{16}+1}{13^{17}+1}< 0\Rightarrow\frac{13^{16}+1}{13^{17}+1}< \frac{13^{16}+\left(1+12\right)}{13^{17}+\left(1+12\right)}\)
\(\Rightarrow B< \frac{13^{16}+13}{13^{17}+13}\\ \Rightarrow B< \frac{13\left(13^{15}+1\right)}{13\left(13^{16}+1\right)}\\ \Rightarrow B< \frac{13^{15}+1}{13^{16}+1}=A\\ \Leftrightarrow A>B\)
Vậy A > B
c. Có: \(\frac{1999^{2000}+1}{1999^{1999}+1}>1\Rightarrow\frac{1999^{2000}+1}{1999^{1999}+1}>\frac{1999^{2000}+\left(1+1998\right)}{1999^{1999}+\left(1+1998\right)}\)
\(\Rightarrow B>\frac{1999^{2000}+1999}{1999^{1999}+1999}\\ \Rightarrow B>\frac{1999\left(1999^{1999}+1\right)}{1999\left(1999^{1998}+1\right)}\\ \Rightarrow B>\frac{1999^{1999}+1}{1999^{1998}+1}=A\\ \Leftrightarrow A< B\)
Vậy A < B
< đó bn
cái đầu thì mẫu hơn tử 1 => cái đầu < 1
cái 2 tử mẫu = nhau => =1
====> cái đầu< cái 2 (nhìn tưởng phức tạp )
đúng nha mk pải off đây
So sánh
\(A=\dfrac{1999^{1999}+1}{1999^{1998}+1}\) ; \(B=\dfrac{1999^{2000}+1}{1999^{1999}+1}\)
Ta có: \(B=\dfrac{1999^{2000}+1}{1999^{1999}+1}>1\) ( vì tử > mẫu )
Do đó: \(B=\dfrac{1999^{2000}+1}{1999^{1999}+1}>\dfrac{1999^{2000}+1+1998}{1999^{1999}+1+1998}=\dfrac{1999^{2000}+1999}{1999^{1999}+1999}=\dfrac{1999.\left(1999^{1999}+1\right)}{1999.\left(1999^{1998}+1\right)}=\dfrac{1999^{1999}+1}{1999^{1998}+1}=A\)
Vậy B > A
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