Áp dụng \(\frac{a}{b}< 1\Rightarrow\frac{a}{b}< \frac{a+c}{b+c}\) (a;b;c \(\in\) N*)

Ta có:

\(B=\frac{10^{20}+1}{10^{21}+1}< \frac{10^{20}+1+9}{10^{21}+1+9}=\frac{10^{20}+10}{10^{21}+10}\)

\(B< \frac{10.\left(10^{19}+1\right)}{10.\left(10^{20}+1\right)}=\frac{10^{19}+1}{10^{20}+1}=A\)

=> A > B

\(B=\frac{10^{20}+1}{10^{21}+1}< 1\)

NÊN \(\frac{10^{20}+1}{10^{21}+1}< \frac{10^{20}+1+9}{10^{21}+1+9}=\frac{10^{20}+10}{10^{21}+10}=\frac{10.\left(10^{19}+1\right)}{10.\left(10^{20}+1\right)}=\frac{10^{19}+1}{10^{20}+1}=A\)

VẬY B<A

Ta thấy B < 1 và 9 > 1 nên ta có:

            B <     10²⁰ + 1 + 9 / 10²¹ + 1 + 9

  =>      B <     10²⁰ + 10 / 10²¹ + 10

  =>      B <     10(10¹⁹ + 1) / 10(10²⁰ + 1)

=>        B <     10¹⁹ + 1 / 10²⁰ + 1 = A

 =>        B < A

Ta thấy:A=\(\frac{10^{19}+1}{10^{20}+1}\)=>10A=\(\frac{10^{20}+10}{10^{20}+1}\)

=>10A=\(\frac{10^{20}+1+9}{10^{20}+1}\)

=>10A=1+\(\frac{9}{10^{20}+1}\)

Ta thấy:B=\(\frac{10^{20}+1}{10^{21}+1}\)

=>10B=\(\frac{10^{21}+10}{10^{21}+1}\)

=>10B=\(\frac{10^{21}+1+9}{10^{21}+1}\)

=>10B=1+\(\frac{9}{10^{21}+1}\)

Do \(\frac{9}{10^{20}+1}\)> \(\frac{9}{10^{21}+1}\)=>A > B

Do \(B=\frac{10^{20}+1}{10^{21}+1}\)<1

\(\Rightarrow B=\frac{10^{20}+1}{10^{21}+1}\)<\(\frac{10^{20}+1+9}{10^{21}+1+9}=\frac{10^{20}+10}{10^{21}+10}=\frac{10.\left(10^{19}+1\right)}{10.\left(10^{20}+1\right)}=\frac{10^{19}+1}{10^{20}+1}=A\)

\(\Rightarrow\)B<A hay A<B

Ta chứng minh bài toán phụ:

Với a<b thì\(\frac{a}{b}< \frac{a+c}{b+c}\)\(\left(c\inℕ^∗\right)\)

Ta có: \(a< b\)

\(\Rightarrow ac< bc\)

\(\Rightarrow ac+ba< bc+ba\)

\(a\left(b+c\right)< b.\left(a+c\right)\)

\(\Rightarrow\frac{a}{b}< \frac{a+c}{b+c}\)

                 đpcm

Áp dụng vào bài toán ta có:

\(\frac{10^{20}+1}{10^{21}+1}< \frac{10^{20}+1+9}{10^{21}+1+9}=\frac{10^{20}+10}{10^{21}+10}=\frac{10.\left(10^{19}+1\right)}{10.\left(10^{20}+1\right)}=\frac{10^{19}+1}{10^{20}+1}\)

Vậy \(\frac{10^{19}+1}{10^{20}+1}>\frac{10^{20}+1}{10^{21}+1}\)

Tham khảo nhé~

Đặt  \(A=\frac{10^{19}+1}{10^{20}+1}\)

\(\Rightarrow10A=\frac{10^{20}+10}{10^{20}+1}=\frac{10^{20}+1+9}{10^{20}+1}=1+\frac{9}{10^{20}+1}\)

\(B=\frac{10^{20}+1}{10^{21}+1}\)

\(\Rightarrow10B=\frac{10^{21}+10}{10^{21}+1}=\frac{10^{21}+1+9}{10^{21}+1}=1+\frac{9}{10^{21}+1}\)

\(\Rightarrow\frac{9}{10^{20}+1}>\frac{9}{10^{21}+1}\)

\(\Rightarrow1+\frac{9}{10^{20}+1}>1+\frac{9}{10^{21}+1}\)

\(\Rightarrow10A>10B\Rightarrow A>B\)

a) Ta có : B = \(\frac{9^{19}+1}{9^{20}+1}\)< \(\frac{9^{19}+1+8}{9^{20}+1+8}\)= \(\frac{9^{19}+9}{9^{20}+9}\)= \(\frac{9\left(9^{18}+1\right)}{9\left(9^{19}+1\right)}\)= \(\frac{9^{18}+1}{9^{19}+1}\)= A

                                                       Vậy A > B

b) Ta có : B = \(\frac{10^{2018}-1}{10^{2019}-1}\)> \(\frac{10^{2018}-1-9}{10^{2019}-1-9}\)= \(\frac{10^{2018}-10}{10^{2019}-10}\)= \(\frac{10\left(10^{2017}-1\right)}{10\left(10^{2018}-1\right)}\)= \(\frac{10^{2017}-1}{10^{2018}-1}\)= A

                                                                         Vậy A < B.

                    NHỚ K CHO MK VỚI NHÉ !!!!!!!!

a A lon hon B

a) 

\(9A=\frac{9^{19}+9}{9^{19}+1}=\frac{9^{19}+1+8}{9^{19}+1}=1+\frac{8}{9^{19}+1}\)

\(9A=\frac{9^{20}+9}{9^{20}+1}=\frac{9^{20}+1+8}{9^{20}+1}=1+\frac{8}{9^{20}+1}\)

ta thấy \(9^{19}+1< 9^{20}+1\Rightarrow\frac{8}{9^{19}+1}>\frac{8}{9^{20}+1}\)

\(\Rightarrow9A>9B\Rightarrow A>B\)