Giải:

a) Gọi dãy đó là A, ta có:

\(A=\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2014}}\) 

\(2A=\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2013}}\) 

\(2A-A=\left(\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2013}}\right)-\left(\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2014}}\right)\) 

\(A=\dfrac{1}{2}-\dfrac{1}{2^{2014}}\) 

Vì \(\dfrac{1}{2}< 1;\dfrac{1}{2^{2014}}< 1\) nên \(\dfrac{1}{2}-\dfrac{1}{2^{2014}}< 1\) 

\(\Rightarrow A< 1\) 

b) \(A=\dfrac{10^{11}-1}{10^{12}-1}\) và \(B=\dfrac{10^{10}+1}{10^{11}+1}\) 

Ta có:

\(A=\dfrac{10^{11}-1}{10^{12}-1}\) 

\(10A=\dfrac{10^{12}-10}{10^{12}-1}\) 

\(10A=\dfrac{10^{12}-1+9}{10^{12}-1}\) 

\(10A=1+\dfrac{9}{10^{12}-1}\) 

Tương tự:

\(B=\dfrac{10^{10}+1}{10^{11}+1}\) 

\(10B=\dfrac{10^{11}+10}{10^{11}+1}\) 

\(10B=\dfrac{10^{11}+1+9}{10^{11}+1}\) 

\(10B=1+\dfrac{9}{10^{11}+1}\) 

Vì \(\dfrac{9}{10^{12}-1}< \dfrac{9}{10^{11}+1}\) nên \(10A< 10B\) 

\(\Rightarrow A< B\)

\(A=\dfrac{10^{11}+1}{10^{12}-1}\)

\(\Rightarrow10A=\dfrac{10^{11}+1}{10^{12}-1}.10\)

\(\Rightarrow10A=\dfrac{10\left(10^{11}+1\right)}{10^{12}-1}\)

\(\Rightarrow10A=\dfrac{10^{12}-10}{10^{12}-1}\)

\(B=\dfrac{10^{10}+1}{10^{11}+1}\)

\(\Rightarrow10B=\dfrac{10^{10}+1}{10^{11}+1}.10\)

\(\Rightarrow10B=\dfrac{\left(10^{10}+1\right).10}{10^{11}+1}\)

\(\Rightarrow10B=\dfrac{10^{11}+10}{10^{11}+1}\)

Ta thấy:

 \(10^{12}-1>10^{12}-10>0\Rightarrow10A< 1\)

\(0< 10^{11}+1< 10^{11}+10\Rightarrow10B>1\)

Mà \(10A< 1;10B>1\)

\(\Rightarrow B>A\).

Bạn tham khảo cách giải này ạ: 

Nếu có 1  phân số \(\dfrac{a}{b}\) < 1 thì a/b < a+n/b+n.

Tương tự ta có: A <   (1011 -1)+11/(1012-1)+10

                        A < 1011+10/1012+10

                        A < 10(1010+1)/10(1011+1)

                        A < 10(1010+1)/10(1011+1)

                        A < 1010+1/1011+1

Vậy A< B ( đcpcm )

 B/A= [(10^10 + 1)/(10^11 + 1)]/[(10^11 - 1)/(10^12 - 1)] 
= [(10^12 - 1).(10^10 + 1)]/[(10^11 - 1).(10^11 + 1)] 
= [(10^22 - 1) + (10^12 - 10^10) ]/((10^22 - 1) 
= 1 + (10^12 - 10^10)/(10^22 - 1) > 1 
=> B > A

 A=10^11-1/10^12-1 < B=10^10+1/10^11=1.

B/A= [(10^10 + 1)/(10^11 + 1)]/[(10^11 - 1)/(10^12 - 1)] 
= [(10^12 - 1).(10^10 + 1)]/[(10^11 - 1).(10^11 + 1)] 
= [(10^22 - 1) + (10^12 - 10^10) ]/((10^22 - 1) 
= 1 + (10^12 - 10^10)/(10^22 - 1) > 1 
=> B > A

mình nghĩ là b

Lời giải:

$B=\frac{10^{11}+10}{10^{12}+10}$

Đặt $10^{11}-1=a; 10^{12}-1=b$ thì $0< a< b$. Khi đó:

$A-B=\frac{a}{b}-\frac{a+11}{b+11}=\frac{11(a-b)}{b(b+11)}<0$

$\Rightarrow A< B$

Dễ vãi

Hỏi 24.10.0.09.98.98888876676.978687877877.9866533145.6655543227.665433346.7646676:2

T

G

A = \(\frac{10^{11}-1}{10^{12}-1}\) và \(B=\frac{10^{10}+1}{10^{11}+1}\)

Đề thế này à

Nếu có 1  phân số a/b < 1 thì a/b < a+n/b+n.

Tương tự ta có: A < (10¹¹ -1)+11/(10¹²-1)+10

                        A < 10¹¹+10/10¹²+10

                        A < 10(10¹⁰+1)/10(10¹¹+1)

                        A < 10(10¹⁰+1)/10(10¹¹+1)

                        A < 10¹⁰+1/10¹¹+1

         Vậy  A < B