Use mathematical induction to prove: 3 is a factor of
n^3 + 2n
Hello
Let P(n) be the property "3 is a factor of $\displaystyle n^3+2n$"
$\displaystyle 1^3+2\:\: 1\;=\;3$ therefore P(1) is true
Let's suppose P(n) true and demonstrate that P(n+1) is true
$\displaystyle (n+1)^3+2(n+1)\;=\;n^3+3n^2+3n+1+2n+2$
$\displaystyle (n+1)^3+2(n+1)\;=\;n^3+2n+3n^2+3n+3$
$\displaystyle (n+1)^3+2(n+1)\;=\;(n^3+2n)+3(n^2+n+1)$
P(n) supposed true => 3 is a factor of $\displaystyle n^3+2n$
3 is clearly a factor of $\displaystyle 3(n^2+n+1)$
Therefore 3 is a factor of $\displaystyle (n^3+2n)+3(n^2+n+1)$
P(n+1) is true