User:Whiteheadj/Square Number

The n+1th square number can be calculated from the nth square number by this formula:

$$n^2 + 2n + 1$$.

This is calculated by:

$$(n+1)^2$$

$$ = (n+1)(n+1)$$

(multiply out the brackets)

$$ = n^2 + n + n + 1^2$$

(simplify)

$$ = n^2 + 2n + 1$$

This can be proved because as it details a way to find the next square number, if it works for $$1^2$$ and for $$2^2$$, then it has to work for all other positive numbers.