Question

Cho S=3⁰+3²+3⁴+...+3²⁰¹⁴

a)Tính tống S

b)Chứng tỏ S chia hết cho 7

Answer 1

Lời giải:

$S=3^0+3^2+3^4+...+3^{2014}$

$3^2S=3^2+3^4+3^6+...+3^{2016}$

$\Rightarrow 3^2S-S=3^{2016}-3^0$

$\Rightarrow 8S=3^{2016}-1$

$\Rightarrow S=\frac{3^{2016}-1}{8}$

b.

$S=(3^0+3^2+3^4)+(3^6+3^8+3^{10})+....+(3^{2010}+3^{2012}+3^{2014})$

$=(1+3^2+3^4)+3^6(1+3^2+3^4)+...+3^{2010}(1+3^2+3^4)$

$=(1+3^2+3^4)(1+3^6+...+3^{2010})=91(1+3^6+...+3^{2010})$

$=7.13(1+3^6+...+3^{2010})\vdots 7$.