[tex]S=1+2\cdot 2+3\cdot 2^2+_{\dots}+2017\cdot 2^{2016}|\cdot 2\\
2S=2+2\cdot 2^2+3\cdot 2^3+_{\dots}+2017\cdot 2^{2017}\\
------------------\\
2S-S=2017\cdot 2^{2017}-2^{2016}-2^{2015}-_{\dots}-2-1\\
S=2017\cdot 2^{2017}-(2^{2016}+2^{2015}+_{\dots}+ 1)\\
*Notam:\\
K=1+2+2^2+_{\dots}+2^{2016}|\cdot 2\\
2K=2+2^2+_{\dots}+2^{2017}\\
-----------\\
2K-K=2^{2017}-1\\
K=2^{2017}-1\\
Revenind:\\
S=2017\cdot 2^{2017}-2^{2017}+1\\
S=2016\cdot 2^{2017}+1\\
[/tex]
[tex]\text{Mai departe aflam ultima cifra a lui S.}\\
u(S)=u (2016)\cdot u(2^{2017})+1\\
u(S)=6\cdot u(2^{2017})+1\\
2^1=2, 2^2=4, 2^3=8, 2^4=..6\\
2017:4=504,r=1\Rightarrow u(2^{2017})=2\\
Asadar:\\
u(S)=6\cdot2+1\\
u(S)=3\\
\text{Un patrat perfect nu poate avea ultima cifra 2,3,7 sau 8.} q.e.d\\
\text{Observatie: Cu u(x) s-a notat ultima cifra a numarului x.}[/tex]
