Question

VA ROOOG MULT
Cum demonstram ca 3^(1)+3^(2)+3^(3)+.....+ 3^(2,016) se divide la 13 ?

Answer 1

[tex]S=3^{1}+3^{2}+3^{3}+.....+3^{2016}\\3^{1}+3^{2}+3^{3}=3+9+27=39=3*13\\ Vom\ grupa\ cate\ 3\ termeni.\\ Vor\ fi\ necesare:\frac{2016}{3}=672\ paranteze\\ S=(3^{1}+3^{2}+3^{3})+....+(3^{2014}+3^{2015}+3^{2016})\\ S=3^{1}*13+3^{4}*13+....+3^{2014}*13\\Dam\ factor\ comun\ 13\\ S=13*(3^{1}+3^{4}+3^{7}+......+3^{2011}+3^{2014})[/tex]