Question

Aratati ca numarul A=1+5+5^2+5^3+......+5^2018 se divide cu 31 va rog Sa ma ajutati chiar Nu stiu si vreau Sa-mi si explicati daca puteti va rooooggggg

Answer 1

   
[tex]\displaystyle\\ A=1+5+5^2+5^3+...+5^{2018}\\ \text{Rescriem suma:}\\ A=5^0+5^1+5^2+5^3+5^4+5^5+\cdots+5^{2016}+5^{2017}+5^{2018}\\ \text{Observam ca suma primilor 3 termeni = 31}\\ 5^0+5^1+5^2=1+5+25=31\\ \text{Trebuie sa grupam termenii sirului in grupe de cate 3 termeni.}\\ \text{Verificam daca putem grupa cate 3.}\\ \text{Analizam exponentii pentru a calcula numarul de termeni.}\\ 0;1;2;3...2018 ~~\Longrightarrow~~1019~\text{termeni}\\ 2019~\vdots~3~=\ \textgreater \ \text{ putem grupa cate 3 termeni.} [/tex]


[tex]\displaystyle\\ A=5^0+5^1+5^2+5^3+5^4+5^5+\cdots+5^{2016}+5^{2017}+5^{2018}\\\\ A=\Big(5^0+5^1+5^2\Big)+\Big(5^3+5^4+5^5\Big)+\cdots+\Big(5^{2016}+5^{2017}+5^{2018}\Big)\\\\ \text{In fiecare grupa dam factor comun.}\\\\ A=\Big(5^0+5^1+5^2\Big)+5^3\Big(5^0+5^1+5^2\Big)+\cdots+5^{2016}\Big(5^0+5^1+5^2\Big)\\\\ \text{Dam factor comun pe }\Big(5^0+5^1+5^2\Big) \\\\ A=\Big(5^0+5^1+5^2\Big)\Big(1+5^3+\cdots+5^{2016}\Big)\\\\ A=31\Big(1+5^3+\cdots+5^{2016}\Big)~\vdots~31\\\\ \boxed{A~\vdots~31} [/tex]