Proof that Σn(n+1) = n(n+1)(n+2)/3 using Mathematical Induction

Put simply, Mathematical Induction is used to prove mathematical statements by the domino effect! That is, if the first domino falls, then the next domino falls, then subsequently, all of the dominoes will fall.

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In this problem, we're required to prove that...

Or, written explicitly...

  for all natural numbers (i.e. n = 1, 2, 3, ...)

Since this is an ordered set, we can use Mathematical Induction to prove this statement to be true. There really are only 2 major steps when in comes to the inductive method...

Step 1: The Basis Test

This is to see if the equation holds true for first number in the set. In this case, n = 1. Thus, on the left-hand side (LHS) we have:

On the right-hand side (RHS) we have:

So this equation satisfies the basis test, in that when n = 1, LHS = RHS = 2. So we're satisfied that the equation holds true so far.

Step 2: The Inductive Test

This is to establish that if the equation holds true for any value in the order (i.e. n = k, where k = 1, 2, 3,...), then the equation holds true for the next value in the order (i.e. n = k +1).

The most convenient (and according to some, the best) way to approach this is to assume that the equation holds true for n = k. So, in our case, this means...

Then, if this assumption is correct, we need to prove that the above holds true for n = k +1. Therefore we need to prove the following...

Now, consider the LHS...

Up to the k(k + 1) term, we have...

...since we assumed the equation holds true for when n = k.

Now...

And thus we have proved: Σn(n + 1) = n(n + 1)(n + 2)/3 for all natural numbers by mathematical induction.

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