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A=
\(\dfrac{10^{101}-1}{10^{102}-1}< \dfrac{10^{101}-1+11}{10^{102}-1+11}=\dfrac{10^{101}+10}{10^{102}+10}=\dfrac{10\left(10^{100}+1\right)}{10\left(10^{101}+1\right)}=B\)
Vậy A<B
CHÚC BẠN HỌC TỐT
Áp dụng a /b > 1 => a/b > a+m/b+m (a;b;m thuộc N*)
Ta có:
\(\frac{100^{10}-1}{100^{10}-3}>\frac{100^{100}-1+2}{100^{10}-3+2}\)
\(>\frac{100^{100}+1}{100^{10}-1}\)
\(A< \frac{\left(10^{10}-1\right)+11}{\left(10^{11}-1\right)+11}< \frac{10^{10}+10}{10^{11}+10}< \frac{10\left(10^9+1\right)}{10\left(10^{10}+1\right)}< \frac{10^9+1}{10^{10}+1}\)
\(\Rightarrow A< B\)
Vậy A<B
Ta có: \(A=\frac{7^{10}}{1+7+7^2+...+7^9}\)
\(\Rightarrow\frac{1}{A}=\frac{1+7+7^2+...+7^9}{7^{10}}=\frac{1}{7^{10}}+\frac{1}{7^9}+\frac{1}{7^8}+...+\frac{1}{7}\)
Lại có: \(B=\frac{5^{10}}{1+5+5^2+...+5^9}\)
\(\Rightarrow\frac{1}{B}=\frac{1+5+5^2+...+5^9}{5^{10}}=\frac{1}{5^{10}}+\frac{1}{5^9}+\frac{1}{5^8}+...+\frac{1}{5}\)
Ta có: \(7^{10}>5^{10}\Rightarrow\frac{1}{7^{10}}< \frac{1}{5^{10}}\)
\(7^9>5^9\Rightarrow\frac{1}{7^9}< \frac{1}{5^9}\)
\(7^8>5^8\Rightarrow\frac{1}{7^8}< \frac{1}{5^8}\)
\(...............................\)
\(7>5\Rightarrow\frac{1}{7}< \frac{1}{5}\)
\(\Rightarrow\frac{1}{7^{10}}+\frac{1}{7^9}+\frac{1}{7^8}+...+\frac{1}{7}< \frac{1}{5^{10}}+\frac{1}{5^9}+\frac{1}{5^8}+...+\frac{1}{5}\)
\(\Rightarrow\frac{1}{A}< \frac{1}{B}\Rightarrow A>B\)
Chúc bạn học tốt !!!
\(B=\frac{10^8}{10^{8-3}}=\frac{10^8}{10^5}=10^3\)
\(A=\frac{10^8+2}{10^8-1}=\frac{10}{10^8-1}+\frac{2}{10^8-1}\)
Vì \(\frac{10}{10^8-1}< 1+\frac{2}{10^8-1}< 1\)
\(\Rightarrow A< B\)
a) A = 1/2.5 + 1/5.8 + 1/8.11 + 1/11.14 + 1/14.17 + 1/17.20
=> 3A = 1/2 - 1/5 + 1/5 - .... + 1/14 - 1/17 + 1/17 - 1/20
=> 3A = 1/2 - 1/20 = 9/20
=> A = 3/20
b) 200410 + 20049 = 20049(1+2004) = 20049 . 2005
200510 = 20059 . 2005
Do 20059 > 20049 nên 200410 + 20049 < 200510
ta có :
\(25^{1008}=\left(5^2\right)^{1008}=5^{2.1008}=5^{2016}\)
mà \(5^{2017}>5^{2016}\)
\(\Rightarrow\)\(5^{2017}>\left(5^2\right)^{1008}\)
\(\Rightarrow\)\(5^{2017}>25^{1008}\)
có \(5^{2017}=\left(5^2\right)^{1008}\times5\)\(=25^{1008}\times5\)
mà \(=25^{1008}\times5\)> \(25^{1008}\)
nên \(5^{2017}>25^{1008}\)