2)Ta có: \(2^{332}< 2^{333}=\left(2^3\right)^{111}=8^{111}\)

              \(3^{223}>3^{222}=\left(3^2\right)^{111}=9^{111}\)

Vì \(8^{111}< 9^{111}\) mà \(2^{332}< 8^{111},3^{223}>9^{111}\) nên suy ra \(2^{332}< 3^{223}\)

Vậy \(2^{332}< 3^{223}\)

1) \(A=\dfrac{10^{2013}+1}{10^{2014}+1}\Rightarrow10A=\dfrac{10^{2014}+10}{10^{2014}+1}=\dfrac{10^{2014}+1}{10^{2014}+1}+\dfrac{9}{10^{2014}+1}=1+\dfrac{9}{10^{2014}+1}\)

\(B=\dfrac{10^{2014}+1}{10^{2015}+1}\Rightarrow10B=\dfrac{10^{2015}+10}{10^{2015}+1}=\dfrac{10^{2015}+1}{10^{2015}+1}+\dfrac{9}{10^{2015}+1}=1+\dfrac{9}{10^{2015}+1}\)Vì: \(10^{2014}+1< 10^{2015}+1\Rightarrow\dfrac{9}{10^{2014}+1}>\dfrac{9}{10^{2015}+1}\Rightarrow1+\dfrac{9}{10^{2014}+1}>1+\dfrac{9}{10^{2015}+1}\)

Nên suy ra \(10A>10B\Rightarrow A>B\)

Gợi ý nhé: bạn hãy so sánh 2014A và 2014B rồi suy ngược lại A và B

Ta có:

2014A=2014²⁰¹⁴+ 2014/2014²⁰¹⁴+1=1+2013/2014²⁰¹⁴+1

2014B=2014²⁰¹³+2014/2014²⁰¹³+1=1+2013/2014²⁰¹³+1

vì 1+2013/2014²⁰¹⁴+1<1+2013/2014²⁰¹³+1 nên 10A < 10B

suy ra A<B

ta có: \(A=\frac{2014^{2013}+1}{2014^{2013}-1}=\frac{2014^{2013}-1+2}{2014^{2013}-1}=1+\frac{2}{2014^{2013}-1}\)

\(B=\frac{2014^{2013}-1}{2014^{2013}-3}=\frac{2014^{2013}-3+2}{2014^{2013}-3}=1+\frac{2}{2014^{2013}-3}\)

\(\Rightarrow\frac{2}{2014^{2013}-1}< \frac{2}{2014^{2013}-3}\)

\(\Rightarrow1+\frac{2}{2014^{2013}-1}< 1+\frac{2}{2014^{2013}-3}\)

=> A < B

Tổng S có 50 phân số

=> S > 1/100 + 1/100 + 1/100 +...+ 1/100 (50 phân số) => S > 1/2.

Vậy S > 1/2

Có \(2004A=\frac{2014^{2015}+2014}{2014^{2015}+1}=\frac{2014^{2015}+1+2013}{2014^{2015}+1}=1+\frac{2013}{2014^{2015}+1}\)

 \(2014B=\frac{2014^{2014}+2014}{2014^{2014}+1}=\frac{2014^{2014}+1+2013}{2014^{2014}+1}=1+\frac{2013}{2014^{2014}+1}\)

 Vì \(\frac{2013}{2014^{2015}+1}< \frac{2013}{2014^{2014}+1}\)

=> \(1+\frac{2013}{2014^{2015}+1}< 1+\frac{2013}{2014^{2014}+1}\)

=> \(A< B\)

SDFGHJI

các cậu trình bày ra 

Tớ thề là \(A>B!!!!!!!!!!!!!!!!!!!!!!!!!!!\)

Mình giải như thế này (giải theo công thức) :

Ta thấy A > 1 

A=\(\frac{2014^{2014}+1}{2014^{2013}+1}\)> \(\frac{2014^{2014}+1+2013}{2014^{2013}+1+2013}\)= \(\frac{2014^{2014}+2014}{2014^{2013}+2014}\)= \(\frac{2014.\left(2014^{2013}+1\right)}{2014.\left(2014^{2012}+1\right)}=\frac{2014^{2013}+1}{2014^{2012}+1}\)

Vậy A>B

\(A=\dfrac{2014^{2013}+1}{2014^{2014}+1}\Leftrightarrow2014A=\dfrac{2014^{2014}+2014}{2014^{2014}+1}=\dfrac{2014^{2014}+1+2013}{2014^{2014}+1}=1+\dfrac{2013}{2014^{2014}+1}\)

\(B=\dfrac{2014^{2012}+1}{2014^{2013}+1}\Leftrightarrow2014B=\dfrac{2014^{2013}+2014}{2014^{2013}+1}=\dfrac{2014^{2013}+1+2013}{2014^{2013}+1}=1+\dfrac{2013}{2014^{2013}+1}\)

Dễ thấy: \(1+\dfrac{2013}{2014^{2014}+1}< 1+\dfrac{2013}{2014^{2013}+1}\) nên \(2014A< 2014B\) hay \(A< B\)