A=3+3²+3³+3⁴+...+3²⁰¹⁶

A=(3+3²)+(3³+3⁴)+...+(3²⁰¹⁵+3²⁰¹⁶)

A=3.(1+3)+3³.(1+3)+...+3²⁰¹⁵.(1+3)

A=3.4+3³.4+...+3²⁰¹⁵.4

A=4.(3+3³+...+3²⁰¹⁵) chia hết cho 4 (đpcm)

A=3+3²+.........+3²⁰¹⁶

A=3.(1+3+9+27)+.....+3²⁰¹³.(1+3+9+27)

A=3.40+.....+3²⁰¹³.40

A=40.(3+...+3²⁰¹³) 

=> A\(⋮40\)

=> ĐPCM .

Câu a:

M = 1/3 - 1/3^2 + 1/3^3 - 1/3^4 + 1/3^5 - 1/3^6 < 1/4

3M = 1 - 1/3 + 1/3^2 - 1/3^3 + 1/3^4 - 1/3^5

3M + M = 3 - 1/3 + 1/3^2 - 1/3^3 + 1/3^4 - 1/3^5 + 1/3 - 1/3^2 + 1/3^3 - 1/3^4 + 1/3^5 - 1/3^6

4M = (1 - 1/3^6) + (-1/3 + 1/3) + (1/3^2 - 1/3^2) + (1/3^4 - 1/3^4) + (1/3^5 - 1/3^5)

4M = 1 - 1/3^6 + 0 + 0+ ..+ 0

M = 1/4 - 1/4.3^6 < 1/4 (đpcm)

4M = 3 - 1/3^6

M = 3/4

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