Question

fie S=1+3+3²+3³+...............+3⁸⁰

demonstrati ca 2S>27²⁷-9

Answer 1

S=1+3+3²+3³+...............+3⁸⁰   inmultim cu 3
3S=3+3²+3³+...............+3⁸⁰+3⁸¹ +1-1              adunam 1 si scadem 1
3S=1+3+3²+3³+...............+3⁸⁰+3⁸¹ -1
3S=S+3⁸¹ -1
2S=3⁸¹ -1
2S=(3³)²⁷ -1
2S=27²⁷ -1    ⇒   2S>27²⁷-9

Answer 2

   
[tex]\displaystyle\\ S = 1+3+3^2+3^3+\cdots +3^{80} = \frac{3^{81}-1}{3-1}=\frac{3^{81}-1}{2}\\\\ 2S = 2\times \frac{3^{81}-1}{2} = \boxed{3^{81}-1}\\\\ 3^{81}-1=3^{3\times 27}-1=(3^3)^{27}-1 =\boxed{27^{27}-1 \ \textgreater \ 27^{27}-9}[/tex]