1. The hexagonal numbers are an increasing sequence of numbers, with each number greater than the one before.

To test whether a number, $x$, is a hexagonal number or not, you can substitute it into the formula:

$\frac{1+\sqrt{8x+1}}{4}$

If the output is a positive integer, then it is a hexagonal number.

A hexagonal number, ${H}_{n}$, and its position value, $n$, satisfy this formula in the following way:

$\frac{1+\sqrt{8{H}_{n}+1}}{4}=n$

By making ${H}_{n}$ the subject, show that the formula for the $n$th term of a hexagonal number is given by

${H}_{n}=2{n}^{2}-n$