Question

a=3+3^2+3^3+......3^2016

Answer 1

$S = 3+ 3^{2} +3^{3} + ... + 3^{2016}|:3 \Rightarrow \\ \\ \Rightarrow \frac{S}{3} = \frac{1}{3}\cdot(3+ 3^{2} +3^{3} + ... + 3^{2016}) \Rightarrow \\ \\ \Rightarrow \frac{S}{3} = 1+3+3^2+...+3^{2015}+3^{2016}-3^{2016} \Rightarrow \\ \\ \Rightarrow \frac{S}{3} = 1+S-3^{2016} \Rightarrow \frac{S}{3}-S = 1- 3^{2016} \Rightarrow \frac{-2S }{3}=1- 3^{2016} \Rightarrow \\ \\ \Rightarrow -2S = 3-3^{2017} \Rightarrow S = - \frac{3-3^{2017}}{2} \Rightarrow S = \frac{3^{2017}-3}{2}$