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

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

\(=5+\frac{1}{1+3+3^2+....+3^8}-2\)  

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

\(\Rightarrow A>B\)

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

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

Tương tự:

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

Vì \(\frac{1}{5}< \frac{1}{3}\) , \(\frac{1}{5^2}< \frac{1}{3^2}\), . . .

nên: \(\frac{1}{\frac{1}{5^9}+\frac{1}{5^8}+...+\frac{1}{5}}>\frac{1}{\frac{1}{3^9}+\frac{1}{3^8}+...+\frac{1}{3}}\)

=> A > B

Vậy đề bạn cho chứng minh A < B là sai nhé.

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

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

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

Ta có:\(B=\frac{1+3+3^2+...+3^9}{1+3+3^2+...+3^8}\)

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

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

vì:\(1+3+3^2+...+3^8< 1+5+5^2+...+5^8\)

Nên A<B(đpcm).

A=1/1+5+5^2+5^3+...+5^8+5+5^2+5^3+...+5^9=1/1+5+5^2+5^3+...+5^8+5.

Tương tự B=1/1+3+3^2+...+3^8+3

=>A>B.

k nha.

=> A>B vì A=1+5+5/1

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

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

Nhận xét: 

\(\frac{1+5+5^2+...+5^8}{5^9}=\frac{1}{5^9}+\frac{1}{5^8}+\frac{1}{5^7}+...+\frac{1}{5}\); \(\frac{1+3+3^2+...+3^8}{3^9}=\frac{1}{3^9}+\frac{1}{3^8}+\frac{1}{3^7}+....+\frac{1}{3}\)

Vì \(\frac{1}{5^9}

Cảm ơn quản lý. Mk cũng bí câu này.

Nhưng còn cách khác dễ hiểu hơn ko?

mk nghĩ là A>B

Gọi tử là : R 

=> \(R=1+5+5^2+5^3+......+5^9\)

\(\Rightarrow5R=5+5^2+5^3+....5^{10}\)

\(\Rightarrow5R-R=5^{10}-1\)

\(\Rightarrow4R=5^{10}-1\)

\(\Rightarrow R=\frac{5^{10}-1}{4}\)

Goij mẫu là M

\(\Rightarrow M=1+5+5^2+5^3+.....+5^8\)

\(\Rightarrow5M=5+5^2+.....+5^9\)

\(\Rightarrow5M-M=5^9-1\)

\(\Rightarrow M=\frac{5^9-1}{4}\)

\(\Rightarrow A=\frac{\frac{5^{10}-1}{4}}{\frac{5^9-1}{4}}=1\)

Tương tự : B

Rồi so sánh thôi dễ mà

Phần B nek :

 Gọi tử là : T

\(\Rightarrow T=1+3+3^2+.....+3^9\)

\(\Rightarrow3T=3+3^2+3^3+.....+3^{10}\)

\(\Rightarrow3T-T=3^{10}-1\)

\(\Rightarrow T=\frac{3^{10}-1}{2}\)

Gọi mẫu là : H

\(\Rightarrow H=1+3+3^2+.....+3^8\)

\(\Rightarrow3H=3+3^2+3^3+.....+3^9\)

\(\Rightarrow3H-H=3^9-1\)

\(\Rightarrow H=\frac{3^9-1}{2}\)

\(\Rightarrow B=\frac{T}{H}=\frac{\frac{3^{10}-1}{2}}{\frac{3^9-1}{2}}=\frac{29524}{9841}=3,0001.....\)

Cho a sửa câu a nha :

 \(\Rightarrow A=\frac{R}{M}=\frac{\frac{5^{10}-1}{4}}{\frac{5^9-1}{4}}=\frac{2441406}{488281}=5,000002048\)

Vậy \(\Rightarrow A>B\left(đpcm\right)\)

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