I was wondering if someone can show me step by step on how to do this question:

Prove that for all integers n >= 1, 6|n (n^2 + 5)

So far I got the bases step for when n = 1.

I am having troubles with the induction step after n = 1.

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- Dec 10th 2011, 02:47 PMxboxweeMathematical Induction
I was wondering if someone can show me step by step on how to do this question:

Prove that for all integers n >= 1, 6|n (n^2 + 5)

So far I got the bases step for when n = 1.

I am having troubles with the induction step after n = 1. - Dec 10th 2011, 02:50 PMProve ItRe: Mathematical Induction
- Dec 10th 2011, 02:52 PMxboxweeRe: Mathematical Induction
Yes.

- Dec 10th 2011, 03:05 PMProve ItRe: Mathematical Induction
The base step is obvious.

For the inductive step, assume that 6 divides $\displaystyle \displaystyle \begin{align*} k(k^2 + 5) \end{align*} $, in other words, write $\displaystyle \displaystyle \begin{align*} k(k^2 + 5) = 6m \end{align*} $ where $\displaystyle \displaystyle \begin{align*} m \end{align*} $ is some other positive integer.

Then we need to show that 6 divides $\displaystyle \displaystyle \begin{align*} (k + 1)\left[(k + 1)^2 + 5\right] \end{align*} $

$\displaystyle \displaystyle \begin{align*} (k + 1)\left[(k + 1)^2 + 5\right] &= (k + 1)\left(k^2 + 2k + 1 + 5\right) \\ &= k\left(k^2 + 5 + 2k + 1\right) + 1\left(k^2 + 5 + 2k + 1\right) \\ &= k\left(k^2 + 5\right) + k(2k + 1) + k^2 + 2k + 6 \\ &= 6m + 2k^2 + k + k^2 + 2k + 6 \\ &= 6m + 3k^2 + 3k + 6 \\ &= 6m + 3k(k + 1) + 6 \\ &= 6m + 3\cdot 2p + 6 \textrm{ where }p \textrm{ is some other positive integer, which we can do since }k(k + 1) \textrm{ is even...} \\ &= 6m + 6p + 6 \\ &= 6\left(m + p + 1\right) \textrm{ which is clearly divisible by }6 \end{align*} $

Q.E.D. - Dec 10th 2011, 03:15 PMxboxweeRe: Mathematical Induction
would this be right?

- Dec 10th 2011, 03:17 PMxboxweeRe: Mathematical Induction
Thanks..

- Dec 10th 2011, 03:27 PMProve ItRe: Mathematical Induction
No it's not a complete proof, as you have not shown that your final line is divisible by 6. In fact, you have not used your assumption at all.

Also, the writing of your base step is atrocious.

$\displaystyle \displaystyle \begin{align*} 6|n\left(n^2 + 5\right) \end{align*} $ is a STATEMENT, that 6 divides $\displaystyle \displaystyle \begin{align*} n\left(n^2 + 5\right) \end{align*} $. It can NOT be used in equations.

You would need to write something like

If $\displaystyle \displaystyle \begin{align*} n = 1 \end{align*} $ then $\displaystyle \displaystyle \begin{align*} n\left(n^2 + 5\right) = 1\left(1^2 + 5\right) = 6 \end{align*} $ which is divisible by 6.