Đặt \(A=\frac15+\frac{2}{5^2}+\cdots+\frac{2016}{5^{2016}}\)

=>\(5A=1+\frac25+\cdots+\frac{2016}{5^{2015}}\)

=>\(5A-A=1+\frac25+\cdots+\frac{2016}{5^{2015}}-\frac15-\frac{2}{5^2}-\cdots-\frac{2016}{5^{2016}}\)

=>\(4A=1+\frac15+\frac{1}{5^2}+\cdots+\frac{1}{5^{2015}}-\frac{2016}{5^{2016}}\)

Đặt \(B=\frac15+\frac{1}{5^2}+\cdots+\frac{1}{5^{2015}}\)

=>\(5B=1+\frac15+\cdots+\frac{1}{5^{2014}}\)

=>\(5B-B=1+\frac15+\cdots+\frac{1}{5^{2014}}-\frac15-\frac{1}{5^2}-\cdots-\frac{1}{5^{2015}}\)

=>\(4B=1-\frac{1}{5^{2015}}=\frac{5^{2015}-1}{5^{2015}}\)

=>\(B=\frac{5^{2015}-1}{4\cdot5^{2015}}\)

TA có: \(4A=1+\frac15+\frac{1}{5^2}+\cdots+\frac{1}{5^{2015}}-\frac{2016}{5^{2016}}\)

\(=1+\frac{5^{2015}-1}{4\cdot5^{2015}}-\frac{2016}{5^{2016}}=1+\frac{5^{2016}-5-8064}{4\cdot5^{2016}}=1+\frac14-\frac{8069}{4\cdot5^{2016}}\)

=>\(4A<1+\frac14=\frac54\)

=>\(A<\frac{5}{16}\)

mà \(\frac{5}{16}<\frac{5}{15}=\frac13\)

nên \(A<\frac13\) (1)

Ta có: \(4A=1+\frac15+\frac{1}{5^2}+\cdots+\frac{1}{5^{2015}}-\frac{2016}{5^{2016}}\)

=>\(20A=5+1+\frac15+\cdots+\frac{1}{5^{2014}}-\frac{2016}{5^{2015}}\)

=>\(20A-4A=5+1+\frac15+\cdots+\frac{1}{5^{2014}}-\frac{2016}{5^{2015}}-1-\frac15-\frac{1}{5^2}-\cdots-\frac{1}{5^{2015}}-\frac{2016}{5^{2016}}\)

=>\(16A=5-\frac{2017}{5^{2015}}-\frac{2016}{5^{2016}}>5\)

=>\(A>\frac{5}{16}\)

=>\(A>\frac{4}{16}=\frac14\) (2)

Từ (1),(2) suy ra 1/4<A<1/3

Bài này dễ,ông không chịu làm thì có ^_^:

Ta có:\(B=1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right)+....+\left(\frac{1}{2^{2014}+1}+....+\frac{1}{2^{2015}}\right)+\frac{1}{2^{2015}+1}+...+\frac{1}{2^{2016}-1}\)

\(>1+\frac{1}{2}+2.\frac{1}{2^2}+2^2.\frac{1}{2^3}+........+2^{2014}.\frac{1}{2^{2015}}\)

\(=1+\frac{1}{2}+\frac{1}{2}+.........+\frac{1}{2}\)  (có 2015 phân số  \(\frac{1}{2}\))

\(=1+2014.\frac{1}{2}+\frac{1}{2}=1008+\frac{1}{2}>1008\)

\(A< \frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+....+\frac{1}{99.100}\)

\(A< \frac{1}{4}-\frac{1}{100}\)

\(A< \frac{6}{25}< \frac{1}{4}\)

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