Bài 1. So sánh A và B biết rằng :
a) A = \(\frac{10^{15}+1}{10^{16}+1}\); B = \(\frac{10^{16}+1}{10^{17}+1}\)
b) A = \(\frac{3}{8^3}\)+\(\frac{7}{8^4}\); B = \(\frac{7}{8^3}\)+\(\frac{3}{8^4}\)
c) A = \(\frac{1}{11}\)+\(\frac{1}{12}\)+\(\frac{1}{13}\)+ ... +\(\frac{1}{19}\)+\(\frac{1}{20}\); B = \(\frac{1}{2}\)
\(a.\)
\(A=\)\(\frac{10^{15}+1}{10^{16}+1}\)
\(10A=\) \(\frac{10\left(10^{15}+1\right)}{10^{16}+1}\)
\(10A=\) \(\frac{10^{16}+10}{10^{16}+1}\)
\(10A=\)\(\frac{10^{16}+1+9}{10^{16}+1}\)
\(10A=\frac{10^{16}+1}{10^{16}+1}+\frac{9}{10^{16}+1}\)
\(10A=1+\frac{9}{10^{16}+1}\)
\(B=\frac{10^{16}+1}{10^{17}+1}\)
\(10B=\frac{10\left(10^{16}+1\right)}{10^{17}+1}\)
\(10B=\frac{10^{17}+10}{10^{17}+1}\)
\(10B=\frac{10^{17}+1+9}{10^{17}+1}\)
\(10B=\frac{10^{17}+1}{10^{17}+1}+\frac{9}{10^{17}+1}\)
\(10B=1+\frac{9}{10^{17}+1}\)
\(\Rightarrow10B< 10A\Rightarrow B< A\)\(\text{( vì tự làm ) }\)
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b
\(A=\frac{3}{8^3}+\frac{7}{8^4}=\frac{3}{8^3}+\frac{3}{8^4}+\frac{4}{8^4}\)
\(B=\frac{3}{8^4}+\frac{7}{8^3}=\frac{3}{8^4}+\frac{3}{8^3}+\frac{4}{8^3}\)
Vì \(\frac{4}{8^4}< \frac{4}{8^3}\)=.> A < B
c
\(A=\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+..+\frac{1}{20}\)
\(\)Ta Thấy :
\(\frac{1}{11}>\frac{1}{20}\)
\(\frac{1}{12}>\frac{1}{20}\)
...
\(\frac{1}{20}=\frac{1}{20}\)
=> A > \(\frac{1}{20}+\frac{1}{20}+...+\frac{1}{20}\)( có 10 phân số \(\frac{1}{20}\))
=> A > \(\frac{1}{20}\).10
=> A > \(\frac{10}{20}\)=\(\frac{1}{2}\)
=> A > \(\frac{1}{2}\)
