Question

Use the Principle of Mathematical Induction to show that the following statement is true for all natural numbers n.

Answer 1

\[P(n)=n ^{3}+2n+6\]

Answer 2

What is the statement..?

Answer 3

\[P(n)=n ^{3}+2n+6\]

Answer 4

is divisible by 3.

Answer 5

ah, there we go. Okay, so lets say P(n) is divisible by three. We seek to show that this implies that P(n+1) is divisible by three.

Answer 6

\[P(n+1) = (n+1)^3 + 2(n+1) + 6 = n^3 + 3n^2 + 3n + 1 + 2n + 2 + 6\]
\[= (n^3+2n+6) + 3n^2 + 3n +3 = P(n) + 3(n^2+n+1)\]
So surely if P(n) is divisible by three, P(n+1) is divisible by three, right?
Now all that remains is to prove it for n = 1.
P(1) = 1 + 2+ 6 = 9 which is divisible by three.
So, by induction, it is true for any natural number n.

Answer 7

Thank you.