Clarification of the 5th Peano

We denote the set of all natural numbers by N. All elements of N are deonted by n.

The 5 Peano Axioms are:

1. 1 is an element of N
2. If n belongs to N, thenits successor n + 1 belongs to N
3. 1 is not the successor to any element in N
4. If n and m, both elements of N, have the same successor, then n = m
5. A subset of N which contains 1, and which contains n + 1 whenever it contains n, must equal N.

Prove that 1^2 + 2^2 +……n^2 = n(n + 1)(2n + 1) / 6, for all n that are elements of N.

Proof: by Mathematical Induction:
We show true for n = 1;
P1: 1^2 = 1(1 + 1)(2*1 + 1) / 6
= 1(2)(3) /6
= 1

Assume the hypothesis is true for Pn:
1^2 + …….+ n^2 = n(n + 1)(2n + 1) / 6

Prove true for Pn+1:
n^2 + (n + 1)^2 = (n +1)[(n + 1) + 1][2(n + 1) + 1] / 6;
therefore,
Pn+1 = (n + 1)(n + 2)(2n + 3) / 6.
Now,
Pn + (n + 1)^2 = n(n + 1)(2n + 1) / 6 + (n + 1)^2
= (n + 1)[n(2n + 1)/6 + (n + 1)
= (n + 1){[ n(2n + 1) + 6(n + 1)] /6}
= (n + 1)[2n^2 + 7n + 6] / 6
= (n + 1)[(2n + 3)(n + 2) / 6]
= (n + 1)(n + 2)(2n + 3) / 6.

QED.