Răspuns :
Răspuns:
Explicație pas cu pas:
Cerința corecta:
"Fie [tex]\bf S=6^1 + 6^2 +6^{3}+6^{4}+6^{5} +....+6^{2016}[/tex]
a) Verifica daca(6 + 6^2) este divizibil 7
b) Arata ca S este divizibil cu 7"
a)
[tex]\bf (6+6^2) = 6 + 36 = 42 =6\cdot7~~\vdots~~7[/tex]
b)
[tex]\bf S=6^1 + 6^2 +6^{3}+6^{4}+6^{5} +....+6^{2016}[/tex]
[tex]\bf S=\Big(6+6^2\Big)+\Big(6^{3}+6^4\Big)+ ...+\Big(6^{2015}+2^{2016}\Big)[/tex]
[tex]\bf S=\Big(6+6^2\Big)+6^{2}\cdot\Big(6^{3-2}+6^{4-2}\Big)+ ...+6^{2014}\cdot\Big(6^{2015-2014}+6^{2016-2014}\Big)[/tex]
[tex]\bf S=\Big(6+6^2\Big)+6^{2}\cdot\Big(6^{1}+6^{2}\Big)+ ...+6^{2014}\cdot\Big(6^{1}+6^{2}\Big)[/tex]
[tex]\bf S=\Big(6+6^2\Big)\cdot\Big(1+6^2+6^4+....+6^{2014}\Big)[/tex]
[tex]\bf S=\Big(6+36\Big)\cdot\Big(1+6^2+6^4+....+6^{2014}\Big)[/tex]
[tex]\bf S=42\cdot\Big(1+6^2+6^4+....+6^{2014}\Big)[/tex]
[tex]\red{\boxed{\bf~ S=6\cdot7\cdot\Big(1+6^2+6^4+....+6^{2014}\Big)~~\vdots~~7~}}[/tex]
[tex]==pav38==[/tex]
Fie S = 6 + 6² + 6³ + ... + 6²⁰¹⁶ .
a) Verifică dacă ( 6 + 6²) ⋮ 7
b) Arată că S ⋮ 7
[tex]\it a)\\ \\ 6+6^2=6+36=42 \ \vdots\ 7\\ \\ b)\\ \\ S=(6+6^2)+6^2(6+6^2)+\ ...\ +6^{2014}(6+6^2)=42(1+6^2+6^4+\ ...\ +6^{2014})\Rightarrow\\ \\ \Rightarrow S\in M_7\Rightarrow S\ \vdots\ 7[/tex]