Question

Câu 19. Cho ${A}=\dfrac{2008^{2008}+1}{2008^{2009}+1}$; ${B}=\dfrac{2008^{2007}+1}{2008^{2008}+1}$. So sánh $A$ và $B$.

Answer 1

Answer 2

Answer 3

Answer 4

ẻtreteteetrttett

Answer 5

Nhận xét: nếu $0<x<y$ và $a>0$ thì $0<ax<ay$ và ��+��<��+��⇒�(�+�)<�(�+�)⇒��<�+��+�xy+ax<xy+ay⇒x(y+a)<y(x+a)⇒yx​<y+ax+a​.

Do đó, (20082008+1)(20082009+1)<(20082008+1)+2007(20082009+1)+2007⇒20082008+120082009+1<2008(20082007+1)2008(20082008+1)(20082009+1)(20082008+1)​<(20082009+1)+2007(20082008+1)+2007​⇒20082009+120082008+1​<2008(20082008+1)2008(20082007+1)​

⇒20082008+120082009+1<20082007+120082008+1⇒20082009+120082008+1​<20082008+120082007+1​.

Vậy $A<B$

Answer 6

Nhận xét: nếu $0<x<y$ và $a>0$ thì $0<ax<ay$ và ��+��<��+��⇒�(�+�)<�(�+�)⇒��<�+��+�xy+ax<xy+ay⇒x(y+a)<y(x+a)⇒yx​<y+ax+a​.

Do đó, (20082008+1)(20082009+1)<(20082008+1)+2007(20082009+1)+2007⇒20082008+120082009+1<2008(20082007+1)2008(20082008+1)(20082009+1)(20082008+1)​<(20082009+1)+2007(20082008+1)+2007​⇒20082009+120082008+1​<2008(20082008+1)2008(20082007+1)​

⇒20082008+120082009+1<20082007+120082008+1⇒20082009+120082008+1​<20082008+120082007+1​.

Vậy $A<B$

Answer 7

Vậy A < B.

Answer 8

x/y < (x + a)/ (y + a) 

(2008²⁰⁰⁸ + 1) / (2008²⁰⁰⁹ + 1 ) <  (2008²⁰⁰⁸ + 1) + 2007 / (2008²⁰⁰⁹ + 1 ) + 2007 => (2008²⁰⁰⁸ + 1) / (2008²⁰⁰⁹ + 1) <  2008 (2008²⁰⁰⁷ + 1) / 2008 ( 2008²⁰⁰⁸ + 1)

=> (2008²⁰⁰⁸ + 1) / (2008²⁰⁰⁹ + 1 )  < (2008²⁰⁰⁷ + 1) / ( 2008²⁰⁰⁸ + 1)

Vậy A < B

Answer 9

A=20082009+120082008+1​

2008⋅A=20082009+200820082009+12008⋅A=20082009+120082009+2008​

=20082009+1+200720082009+1=20082009+120082009+1+2007​

=1+200720082009+1=1+20082009+12007​

B=20082007+120082008+1B=20082008+120082007+1​

2008⋅B=20082008+200820082008+12008⋅B=20082008+120082008+2008​

=20082008+1+200720082008+1=20082008+120082008+1+2007​

=1+200720082008+1=1+20082008+12007​

Ta có: 20082009+1>20082008+120082009+1>20082008+1

$\Rightarrow$

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Answer 10

A=20082009+120082008+1​

2008⋅A=20082009+200820082009+12008⋅A=20082009+120082009+2008​

=20082009+1+200720082009+1=20082009+120082009+1+2007​

=1+200720082009+1=1+20082009+12007​

B=20082007+120082008+1B=20082008+120082007+1​

2008⋅B=20082008+200820082008+12008⋅B=20082008+120082008+2008​

=20082008+1+200720082008+1=20082008+120082008+1+2007​

=1+200720082008+1=1+20082008+12007​

Ta có: 20082009+1>20082008+120082009+1>20082008+1

$\Rightarrow$

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Answer 11

A=20082009+120082008+1​

2008⋅A=20082009+200820082009+12008⋅A=20082009+120082009+2008​

=20082009+1+200720082009+1=20082009+120082009+1+2007​

=1+200720082009+1=1+20082009+12007​

B=20082007+120082008+1B=20082008+120082007+1​

2008⋅B=20082008+200820082008+12008⋅B=20082008+120082008+2008​

=20082008+1+200720082008+1=20082008+120082008+1+2007​

=1+200720082008+1=1+20082008+12007​

Ta có: 20082009+1>20082008+120082009+1>20082008+1

$\Rightarrow$

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Answer 12

nếu 0<x<y và a>0 thì 0<ax<ay và xy + ay ⇒ x(y+a)<y(x+a)⇒y< x+a / y+a
do đó:( 2008²⁰⁰⁸+1)/(2008²⁰⁰⁹+1)<(2008²⁰⁰⁸+1)+2007/(2008²⁰⁰⁹+1)+2007⇒2008²⁰⁰⁸+11/2008²⁰⁰⁹+1
<2008(2008²⁰⁰⁷+1)/2008(2008²⁰⁰⁸+1)⇒2008²⁰⁰⁸+1/2008²⁰⁰⁹+1< 2008²⁰⁰⁷+1/2008²⁰⁰⁸+1.
Vậy A < B.

Answer 13

A=2008²⁰⁰⁸+1/2008²⁰⁰⁹+1

2008A=2008²⁰⁰⁹+2008/2008²⁰⁰⁹+1

2008A=2008²⁰⁰⁹+1+2007/2008²⁰⁰⁹+1

2008A=1 +2007/2008²⁰⁰⁹+1

B=2008²⁰⁰⁷+1/2008²⁰⁰⁸+1

2008B=2008²⁰⁰⁸+2008/2008²⁰⁰⁸+1

2008B=2008²⁰⁰⁸+1+2007/2008²⁰⁰⁸+1

2008B=1 +2007/2008²⁰⁰⁸+1

1 +2007/2008²⁰⁰⁹+1<1 +2007/2008²⁰⁰⁸+1 

=>B>A

Answer 14

A=1,B=2

Suy ra B>A

Answer 15

A<B

 

Answer 16

A=20082009+120082008+1​

2008⋅A=20082009+200820082009+12008⋅A=20082009+120082009+2008​

=20082009+1+200720082009+1=20082009+120082009+1+2007​

=1+200720082009+1=1+20082009+12007​

B=20082007+120082008+1B=20082008+120082007+1​

2008⋅B=20082008+200820082008+12008⋅B=20082008+120082008+2008​

=20082008+1+200720082008+1=20082008+120082008+1+2007​

$=$

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Answer 17

A nhỏ hơn B