Question

S=3+3la puterea2+3la puterea 3....+3la puterea12

a)aratati ca suma S este divizibila cu 12
b)aratati ca suma S este divizibila cu 13
c)aratati ca suma S este divizibila cu 5

Answer 1

[tex]a)S=(3+3^2)+3^{2}(3+3^2)+...+3^{10}(3+3^2)\\ S=(3+3^2)(1+3^2+...+3^{10})\\ S=12(1+3^2+...+3^{10})\\ .................................................................................\\ b)S=3(1+3+3^2)+3^{4}(1+3+3^2)+...+3^{10}(1+3+3^2)\\ S=(1+3+3^2)(3+3^4+...+3^{10})\\ S=13( 3+3^4+...+3^{10})\\ ...............................................................................\\ c)S=(3+3^2+3^3+3^4)+3^4(3+3^2+3^3+3^4)+3^8(3+3^2+3^3+3^4)\\ S=(3+3^2+3^3+3^4)(1+3^4+3^8)\\ S=(3+9+27+81)(1+3^4+3^8)\\ [/tex]
[tex]S=120\cdot (1+3^4+3^8)\\ S=5\cdot 24\cdot (1+3^4+3^8)[/tex]