mk nghĩ là A>B

\(A=\frac{1+5+5^2+...+5^9}{1+5+5^2+...+5^8}=\frac{1+5\left(1 +5+5^2+...+5^8\right)}{1+5+5^2+...+5^8}=5+\frac{1}{1+5+5^2+...+5^8} \)

\(B=\frac{1+3+3^2+....+3^9}{1+3+3^2+....+3^8}=\frac{1+3\left(1+3+3^2+....+3^8\right)}{1+3+3^2+....+3^8}=3+\frac{1}{1+3+3^2+....+3^8}\)

\(=5+\frac{1}{1+3+3^2+....+3^8}-2\)  

Có: \(\frac{1}{1+5+5^2+...+5^8}>0\)              và      \(\frac{1}{1+3+3^2+....+3^8}-2< 0\)

\(\Rightarrow A>B\)

A=1+5+5^2+..+5^9/1+5+5^2+...+5^8

=1+5^9/1+5+5^2+...+5^8 

B=1+3+3^2+..+3^9/1+3+3^2+..+3^8

=1+3^9/1+3+3^2+..+3^8

đặt A' =1+5+5^2+...+5^8

5A'=5+5^2+5^3+...+5^9

5A'-A'=5+5^2+5^3+...+5^9-5-1-5-5^2-...-5^8

4A'=5^9-1=>A'=(5^9-1):4

tương tự B'=(3^9-1):4

A=1+5^9/(5^9-1)/4=4.5^9/5^9-1

B=1+3^9/(3^9-1)/4=4.3^9/3^9-1

=> A<B

Ta có: \(5\left(1+5+5^2+...+5^9\right)-\left(1+5+5^2+...+5^9\right)\)

= \(\left(5+5^2+5^3+...+5^{10}\right)-\left(1+5+5^2+...+5^9\right)\)

\(4\left(1+5+5^2+...+5^9\right)\)\(=5^{10}-1\)

=> \(1+5+5^2+...+5^9=\frac{5^{10}-1}{4}\)

Tương tự: \(1+5+5^2+....+5^8=\frac{5^9-1}{4}\)

=> \(A=\frac{\frac{5^{10}-1}{4}}{\frac{5^9-1}{4}}=\frac{5^{10}-1}{5^9-1}=\frac{5\left(5^9-1\right)+4}{5^9-1}=5+\frac{4}{5^9-1}>5\)

Tương tự:

\(1+3+3^2+...+3^9=\frac{3^{10}-1}{2}\)

và \(1+3+3^2+...+3^8=\frac{3^9-1}{2}\)

=>\(B=\frac{3^{10}-1}{3^9-1}=\frac{3\left(3^9-1\right)+2}{3^9-1}=3+\frac{2}{3^9-1}< 5\)

=> A >  5 > B

A= \(\frac{1+5+5^2+...+5^9}{1+5+5^2+...+5^8}\)

  = \(\frac{1}{1+5+5^2+...+5^8}+\frac{5\left(1+5+5^2+...+5^8\right)}{1+5+5^2+...+5^8}\)

mà \(\frac{1}{1+5+5^2+...+5^8}\approx0\)

suy ra: A= 5.

chứng minh tương tự, ta có: B=3

5 > 3 --> A>B

kieu nay la ko tinh ra ket qua hay so sanh

A=1+C; voi C=5^9/(1+...5^8)=1/(1/5^9+1/5^8+...+1/5)

B=1+D;voi D=3^9/(1+..3^8)=1/(1/3^9+1/3^8+...+1/3)

C=1/E; voi E=(1/5^9+1/5^8+...+1/5)

D=1/f; voi F=(1/3^9+1/3^8+...+1/3)

=> F-E=(1/3-1/5)+...+(1/3^9-1/5^9) >0=> F>E

=> C>D=> A>B