Well, what's your base case going to be?
So I started doing this proof then realized I hasn't really used induction. I said that n^3 + 5n is equal to n^3 - n + 6n = n(n^2 - 1) + 6n = n(n-1)(n+1) + 6n = (n-1)(n)(n+1) + 6n
Now I know that 6n is divisible by 6 and the product of 3 consecutive integers is divisible by 6 so that the entire term is divisble by 6 but I don't know how to write a proper proof using that explanation. I know you can do this using induction although im not sure how. Thanks for your help
P(k)
The proposition for n=k is
is divisible by 6
P(k+1)
The proposition for the "next n" is
is divisible by 6.
You need to prove that the Induction Domino-Effect exists
by showing that P(k+1) will be true "if" P(k) is true.
Hence, either write P(k+1) in terms of P(k),
when we will see if P(k+1) will be true if P(k) is..
Or work towards P(k+1) from P(k).
Proof
Since k and k+1 are a pair of consecutive even and odd terms, is a multiple of 6 for
Then we can see if P(k+1) will be true "if" P(k) is.
Prove for a base case to topple the dominoes.