\(5A=\frac{5^{2011}+5}{5^{2011}+1}=1+\frac{4}{5^{2011}+1}\)

\(5B=\frac{5^{2010}+5}{5^{2010}+1}=1+\frac{4}{5^{2010}+1}\)

\(5B>5A\Rightarrow B>A\)

Ta có:

A = \(\frac{5^{2010}+1}{5^{2011}+1}\)

5A = \(\frac{5^{2011}+5}{5^{2011}+1}\) = \(\frac{5^{2011}+1+4}{5^{2011}+1}\) = 1 + \(\frac{4}{5^{2011}+1}\)

B = \(\frac{5^{2009}+1}{5^{2010}+1}\)

5B = \(\frac{5^{2010}+5}{5^{2010}+1}\) = \(\frac{5^{2010}+1+4}{5^{2010}+1}\) = 1 + \(\frac{4}{5^{2010}+1}\)

Vì 1 + \(\frac{4}{5^{2011}+1}\) < \(\frac{4}{5^{2010}+1}\) => 5A < 5B

Vì 5A < 5B => A < B

Ta có: \(5A=\frac{5^{2011}+5}{5^{2011}+1}=\frac{5^{2011}+1+4}{5^{2011}+1}=1+\frac{4}{5^{2011}+16}\)

\(5B=\frac{5^{2010}+5}{5^{2010}+1}=\frac{5^{2010}+1+4}{5^{2010}+1}=1+\frac{4}{5^{2010}+1}\)

Vì \(\frac{4}{5^{2011}+1}< \frac{4}{5^{2010}+1}\Rightarrow5A< 5B\Rightarrow A< B\)

Ta có:

A = \(\frac{5^{2010}+1}{5^{2011}+1}\)

\(\Rightarrow5A=\frac{5.\left(5^{2010}+1\right)}{5^{2011}+1}\)\(=\frac{5^{2011}+5}{5^{2011}+1}=1+\frac{4}{5^{2011}+1}\)

B=\(\frac{5^{2009}+1}{5^{2010}+1}\)

\(\Rightarrow5B=\frac{5.\left(5^{2009}+1\right)}{5^{2010}+1}=\frac{5^{2010}+5}{5^{2010}+1}=1+\frac{4}{5^{2010}+1}\)

Ta thấy \(5^{2011}+1>5^{2010}+1\)

\(\Rightarrow\frac{4}{5^{2011}+1}< \frac{4}{5^{2010}+1}\)

\(\Rightarrow1+\frac{4}{5^{2011}+1}< 1+\frac{4}{5^{2010}+1}\)

Hay 5.A<5.B

Vậy A<B (đpcm)

Ta có :

\(B=\frac{5^{2009}+1}{5^{2010}+1}=\frac{\left(5^{2009}+1\right).10}{\left(5^{2010}+1\right).10}=\frac{5^{2010}+10}{5^{2011}+10}\)

Ta thấy :

\(5^{2010}=5^{2010};1< 10\Rightarrow5^{2010}+1< 5^{2010}+10\)

\(5^{2011}=5^{2011};1< 10\Rightarrow5^{2011}+1< 5^{2011}+10\)

Suy ra : \(A< B\)

Vậy \(A< B\)

\(A< 1\)

\(A< \frac{5^{2010}+1}{5^{2011}+1}\)

\(A< \frac{5^{2010}+1+4}{5^{2011}+1+4}\)

\(A< \frac{5^{2010}+5}{5^{2011}+5}\)

\(A< \frac{5\left(5^{2009}+1\right)}{5\left(5^{2010}+1\right)}\)

\(A< \frac{5^{2009}+1}{5^{2010}+1}\)

\(A< B\)

Đầu tiên chúng ta sẽ so sánh như sau

5^2010 và 5^2009

vì 2010>2009 nên 5^2010>5^200 (1)

1/5^2011+1 và 1/5^2010+1

vì 2011+1=2012

   2010+1=2011

mà 2012>2011 nên 1/5^2011+1>1/5^2010+1 (2)

Từ 1 và 2 ta có thể suy ra A>B

Vậy A>B

ta có 2010 >2009 suy ra 5^2010 >5^2009 suy ra 5^2010 + 1>5^2009 +1                                               (1)

         2011>2010 suy ra 5^2011 >5^2010 suy ra 1/5^2011<1/5^2010 suy ra 1/5^2011 +1 <1/5^2010 + 1  (2)

từ (1) và (2) => A=B

A = \(1+\frac{9^{2010}}{1+9+9^2+....+9^{2009}}\)= \(1+1:\frac{1+9+9^2+....+9^{2009}}{9^{2010}}\)= \(1+1:\left(\frac{1}{9^{2010}}+\frac{1}{9^{2009}}+\frac{1}{9^{2008}}+...+\frac{1}{9}\right)\)

B = \(1+\frac{5^{2010}}{1+5+5^2+....+5^{2009}}\)= \(1+1:\frac{1+5+5^2+...+5^{2009}}{5^{2010}}\)= \(1+1:\left(\frac{1}{5^{2010}}+\frac{1}{5^{2009}}+...+\frac{1}{5}\right)\)

Do \(\frac{1}{9^{2010}}

có đúng đề không vậy 

Đặt M = \(1+9+9^2+......+9^{2010}\)

\(9M=9+9^2+9^3+......+9^{2011}\)

\(9M-M=8M=9^{2011}-1\)

Đặt K = \(1+9+9^2+......+9^{2009}\)

\(9K=9+9^2+9^3+.....+9^{2010}\)

\(9K-K=8K=9^{2010}-1\)

\(\Rightarrow A=\frac{9^{2011}-1}{9^{2010}-1}\)

Đặt H=\(1+5+5^2+....+5^{2010}\)

\(5H=5+5^2+......+5^{2011}\)

\(5H-H=4H=5^{2011}-1\)

ĐẶT G = \(1+5+5^2+.......+5^{2009}\)

\(5G-G=4G=5^{2010}-1\)

\(\Rightarrow B=\frac{5^{2011}-1}{5^{2010}-1}\)

Rồi bạn so sánh sẽ ra ngay

áp dụng tc \(\frac{a}{b}< 1\Rightarrow\frac{a+m}{a+m}< 1\left(m\in N\right)\)

Ta có: \(A=\frac{5^{2010}+1}{5^{2011}+1}< \frac{5^{2010}+1+4}{5^{2011}+1+4}\)\(=\frac{5^{2010}+5}{5^{2011}+5}=\frac{5.\left(5^{2009}+1\right)}{5.\left(5^{2010}+1\right)}=\frac{5^{2009}+1}{5^{2010}+1}\)

\(\Rightarrow A< B\)

#)Giải : 

Đầu tiên ta so sánh : 

5²⁰¹⁰ và 5²⁰⁰⁹ 

Vì 2010 > 2009 => 5²⁰¹⁰ > 5²⁰⁰⁹    (1)

Tiếp theo :

1/5^{2011 + 1} và 1/5^{2010 + 1}

Vì 2011 + 1 = 2012 và 2010 + 1 = 2011 

Mà 2012 > 2011 => 1/5^{2011 + 1} > 1/5^{2010 + 1}   (2)

Từ (1) và (2) => 5²⁰¹⁰ + 1/5²⁰¹¹⁺¹ > 5²⁰⁰⁹+1/5²⁰¹⁰⁺¹ => A > B

Vậy : A > B

#)Nếu đúng thì bn bảo mk nha :D

      #~Will~be~Pens~#

Hoàng Nguyên Hiếu sai rồi 

Giúp mk đi!!!!! Help me!!!!!!!!!

ta có:5A = \(\frac{5^{2011}+5}{5^{2011}+1}\) = 1+\(\frac{4}{5^{2011}+1}\)

    5B=\(\frac{5^{2010}+5}{5^{2010}+1}\)=1+\(\frac{4}{5^{2010}+1}\)

 \(\frac{4}{5^{2011}+1}\)<\(\frac{4}{5^{2010}+1}\)=>1+\(\frac{4}{5^{2011}+1}\)<1+\(\frac{4}{5^{2010}+1}\)

=>5A<5B=>A<B

vậy:A<B

chúc pn hok tốt ^_^

Sorry mình viết thiếu B=5 mũ 2009\5 mũ 2010+1