Proof that n^3 - n is divisible by 3 using Mathematical Induction

Using mathematical induction prove that...

n³ - n is divisible by 3 ∀ n ≥ 2

Mathematical induction is quite a mechanical process and there's really only three steps you need to remember.

1. The first step is to verify that the statement holds true for a special case of n. Normally it is n = 1, but in this case, we will choose n = 2 because n doesn't go any lower than that.
2. The next step is to verify that this statement holds true for any value of n
3. Then by induction we that this statement holds true for all positive values of n ≥ 2, or whichever condition you may have.

So the first step, we test if it is true for n = 2. So...

2³ - 2 = 8 - 2 = 6 = 3 x 2

Now it is obvious that it is divisible by 3, so this statement holds true for n = 2.

The next step, we assume that it is true for n is equal to any positive integer, k. So we write...

k³ - k = 3J where J = 1,2,3...

And for the third step, we prove that it is true for the next value of n, or n = k + 1.

Please watch the video for the rest of the proof! And don't hesitate to ask me any questions using the comments and best of luck!
Here are my formulas

1+sin(x)^2+3

• x^2+y^2+z^2
• a^2+b^2