Question

Aratati ca numerele urmatoare sunt compuse pentru orice valoare a numarului natural n:
a) A=6 la puterea n + 3 la puterea n + 2 la puterea n+ 1
b) B= 15 la puterea n + 5 la puterea n+1 + 3 la puterea n +5

Answer 1

$a)~A=6^{n}+3^{n}+2^{n}+1$

$A=(2*3)^{n}+3^{n}+2^{n}+1$

$A=2^{n}*3^{n}+1*3^{n}+2^{n}+1$

$A=3^{n}(2^{n}+1)+1*(2^{n}+1)$

$A=(2^{n}+1)(3^{n}+1)$

$=>A~este~compus$


$b)~B=15^{n}+5^{n+1}+3^{n}+5$

$B=(3*5)^{n}+5^{n}*5+3^{n}+5$

$B=3^{n}*5^{n}+5^{n}*5+3^{n}+5$

$B=3^{n}*5^{n}+3^{n}*1+5*5^{n}+5*1$

$B=3^{n}(5^{n}+1)+5(5^{n}+1)$

$B=(5^{n}+1)(3^{n}+5)$

$=>B~este~compus$