Math Induction: Solving (n^3+5n)=6q

[kring_c14]

Homework Statement

n $$\epsilon$$$$N$$=n$$\geq0$$
6 divides (n$$^{3}$$+5n)

Homework Equations

(n$$^{3}$$+5n)=6q

The Attempt at a Solution

by expanding and simplifying and later on substituting 6q in
(n+1)$$^{}3$$+5(n+1)

ive arrived at 6q+3n$$^{}2$$ +3n+6

then.. I am stuck...pls help...lot of thanks!

 

[kring_c14]

i tried multiplying and dividing it by two so that the 6 would be factored out...ive got fractions..its supposed to be whole numbers

 

[Anadyne]

(n+1)^3 +5(n+1)
n^3 + 3n^2 + 3n + 1 + 5n + 5
(n^3 + 5n) + 3n^2 + 3n + 6
6q + 3n^2 + 3n + 6
This reduces the problem down to proving that 3n^2 + 3n + 6 is a multiple of 6

3(n^2 + n + 2)
Now you just have to prove that n^2 + n + 2 is a multiple of 2, (is even).