Question

Aratati ca nr. 3 +3²+3³+3⁴+.............+3 la 2014 este divizibil cu 4 .

Answer 1

$S=3+3^{2}+3^{3}+3^{4}+....+3^{2014}|*3\\ 3S=3*(3+3^{2}+3^{3}+3^{4}+....+3^{2014})\\ 3S=3*3+3*3^{2}+3*3^{3}+3*3^{4}+.....+3*3^{2014}\\ 3S=3^{1+1}+3^{2+1}+3^{3+1}+3^{4+1}+......+3^{2014+1}\\ 3S=3^{2}+3^{3}+3^{4}+3^{5}+......+3^{2015}\\ 3S=(3+3^{2}+3^{3}+3^{4}+......+3^{2014})+3^{2015}-3\\3S=S+3^{2015}-3\\ 3S-S=3^{2015}-3\\2S=3^{2015}-3\\S=\frac{3^{2015}-3}{2}\\ U(3^{2015})=U(3^{503*4+3})=U(3^{3})=27\\\frac{27-3}{2}=12\\ Ultimele\ doua\ cifre\ ale\ numarului\ sunt\ egale\ cu\ 12\\ S\ divizbila\ cu\ 4$