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8 tháng 6 2018

\(\dfrac{1+2+2^2+2^3+...+2^{2008}}{1-2^{2009}}=\dfrac{\left(2-1\right).\left(1+2+2^2+2^3+...+2^{2008}\right)}{1-2^{2009}}=\dfrac{2^{2009}-1}{1-2^{2009}}=-1\)

3 tháng 8 2023

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

Chúc bạn học tốt

14 tháng 8 2019

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

16 tháng 10 2022

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

12 tháng 4 2018
https://i.imgur.com/XM85QHj.jpg
18 tháng 3 2017

Bài 1:

a) Ta có: \(13A=\dfrac{13^{16}+13}{13^{16}+1}=1+\dfrac{12}{13^{16}+1}\)

\(13B=\dfrac{13^{17}+13}{13^{17}+1}=1+\dfrac{12}{13^{17}+1}\)

\(\dfrac{12}{13^{16}+1}>\dfrac{12}{13^{17}+1}\Rightarrow1+\dfrac{12}{13^{16}+1}>1+\dfrac{12}{13^{17}+1}\)

\(\Rightarrow13A>13B\)

\(\Rightarrow A>B\)

Vậy A > B

b) Ta có: \(1999C=\dfrac{1999^{2000}+1999}{1999^{2000}+1}=1+\dfrac{1998}{1999^{2000}+1}\)

\(1999D=\dfrac{1999^{1999}+1999}{1999^{1999}+1}=1+\dfrac{1998}{1999^{1999}+1}\)

\(\dfrac{1998}{1999^{2000}+1}< \dfrac{1998}{1999^{1999}+1}\Rightarrow1+\dfrac{1998}{1999^{2000}+1}< 1+\dfrac{1999}{1999^{1999}+1}\)

\(\Rightarrow1999C< 1999D\)

\(\Rightarrow C< D\)

Vậy C < D

18 tháng 1 2019

Ta có:

\(A-B=\dfrac{1999^{1999}+1}{1999^{1998}+1}-\dfrac{1999^{2000}+1}{1999^{1999}+1}\)

\(=\dfrac{\left(1999^{1999}+1\right)^2-\left(1999^{1998}+1\right)\left(1999^{2000}+1\right)}{\left(1999^{1998}+1\right)\left(1999^{1999}+1\right)}\)

\(=\dfrac{1999^{3998}+2\cdot1999^{1999}+1-\left(1999^{3998}+1999^{1998}+1999^{2000}+1\right)}{1999^{3997}+1999^{1998}+1999^{1999}+1}\)

\(=\dfrac{2\cdot1999^{1999}-1999^{1998}-1999^{2000}}{1999^{3997}+1999^{1998}+1999^{1999}+1}\)

\(2\cdot1999^{1999}-1999^{1998}-1999^{2000}=-\left[\left(1999^{999}\right)^2-2\cdot1999^{999}\cdot1999^{1000}+\left(1999^{1000}\right)^2\right]\)

\(=-\left(1999^{999}-1999^{1000}\right)^2< 0\)

Mà mẫu số > 0

\(\Rightarrow A-B< 0\Leftrightarrow A< B\)

28 tháng 1 2019

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

12 tháng 8 2018

1,

x+1/2+x+1/3+x+1/4-x+1/5-x+1/6=0

(x+1)(1/2+1/3+1/4-1/5-1/6)=0

vì 1/2+1/3+1/4-1/5-1/6 khác 0

suy ra x+1=0 suy ra x=-1

15 tháng 7 2018

\(\left(1+2+3+...+100\right).\left(1^2+2^2+3^3+...+100^2\right).\left(65.111-13.15.37\right)\)

\(=\left(1+2+3+...+100\right).\left(1^2+2^2+3^3+...+100^2\right).\left(7215-7215\right)\)

\(=\left(1+2+3+...+100\right).\left(1^2+2^2+3^3+...+100^2\right).0\)

\(=0\)

\(1999.1999.1998-1998.1998.1999\)

\(=1999.1998.\left(1999-1998\right)\)

\(=1999.1998.1\)

Tham khảo nhé~

15 tháng 7 2018

13453 nhe

5 tháng 4 2021

A ) Đặt 

\(A=0,1+0,2+...+1,9\\ \Rightarrow10A=1+2+3+..+19\\ =\left(1+19\right)\cdot\dfrac{19}{2}\\ =20\cdot\dfrac{19}{2}\\ =10\cdot19=190\\ \Rightarrow A=19\)

b) \(\left(1999\cdot1998+1998\cdot1997\right)\cdot\left(1+\dfrac{1}{2}:1\dfrac{1}{2}-1\dfrac{1}{3}\right)\)

\(=1998\cdot\left(1999+1997\right)\cdot\left(1+\dfrac{1}{2}:\dfrac{3}{2}-\dfrac{4}{3}\right)\)

\(=1998\cdot3996\cdot\left(1+\dfrac{1}{3}-\dfrac{4}{3}\right)\)

\(=1998\cdot3996\cdot0=0\)

29 tháng 10 2018

\(P=\sqrt{1+1999^2+\dfrac{1999^2}{2000^2}}+\dfrac{1999}{2000}=\sqrt{\dfrac{2000^2+1999^2.2000^2+1999^2}{2000^2}}+\dfrac{1999}{2000}=\dfrac{\sqrt{2000^2+\left(2000-1\right)^2.2000^2+1999^2}}{2000}+\dfrac{1999}{2000}=\dfrac{\sqrt{2000^2+\left(2000^2-2.2000+1\right).2000^2+1999^2}+1999}{2000}=\dfrac{\sqrt{2000^2+2000^4-2.2000.2000^2+2000^2+1999^2}+1999}{2000}=\dfrac{\sqrt{2000^4+2.2000^2-2.\left(1999+1\right).2000^2+1999^2}+1999}{2000}=\dfrac{\sqrt{2000^4+2.2000^2-2.1999.2000^2-2.2000^2+1999^2}+1999}{2000}=\dfrac{\sqrt{2000^4-2.1999.2000^2+1999^2}+1999}{2000}=\dfrac{\sqrt{\left(2000^2-1999\right)^2}+1999}{2000}=\dfrac{2000^2-1999+1999}{2000}=\dfrac{2000^2}{2000}=2000\)