Question

Cho biểu thức: S=3⁰+ 3¹+3²+......+3²⁰¹⁷

a)Chứng tỏ rằng :S không chia hết cho 3 ; \(S⋮4\)

b) Rút gọn S

Answer 1

a/ Ta có :

\(S=1+3+3^2+........+3^{2017}\)

\(\Leftrightarrow S=\left(1+3\right)+\left(3^2+3^3\right)+......+\left(3^{2016}+3^{2017}\right)\)

\(\Leftrightarrow S=1\left(1+3\right)+3^2\left(1+3\right)+......+3^{2016}\left(1+3\right)\)

\(\Leftrightarrow S=1.4+3^2.4+........+3^{2016}.4\)

\(\Leftrightarrow S=4\left(1+3^2+......+3^{2016}\right)⋮4\left(đpcm\right)\)

b/ \(S=1+3+..........+3^{2017}\)

\(\Leftrightarrow3S=3+3^2+.........+3^{2017}+3^{2018}\)

\(\Leftrightarrow3S-S=\left(3+3^2+..........+3^{2018}\right)-\left(1+3+.....+3^{2017}\right)\)

\(\Leftrightarrow2S=3^{2018}-1\)

\(\Leftrightarrow S=\dfrac{3^{2018}-1}{2}\)