Question

Aratati ca numarul n=5+ 5²+5³+...+5${2016}$ , este divizibil cu 780.
Va rog frumos!

Answer 1

[tex]S=5+5^{2}+5^{3}+.....+5^{2016}\\5+5^{2}+5^{3}+5^{4}=5+25+125+625=780\\ Vom\ grupa\ cate\ 4\ trmeni.\\ Vor\ fi\ necesare:\frac{2016}{4}=504\ paranteze\\ S=(5+5^{2}+5^{3}+5^{4})+.....+(5^{2013}+5^{2014}+5^{2015}+5^{2016})\\ S=780*5^{0}+780*5^{4}+....+780*5^{2012}\\Dam\ factor\ comun\ 780:\\ S=780*(5^{0}+5^{4}+5^{8}+....+5^{2008}+5^{2012})[/tex]