This question has 3 part(s).

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

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

This question has 3 part(s).

This investigation introduces and explores the properties and patterns of the sequence of hexagonal numbers.

Hexagonal numbers are numbers formed by summing equally spaced dots around hexagons that sit inside of each other, as shown below.

If $n$ is the position of the term in the sequence then ${H}_{n}$ is the $n$th hexagonal number.

By counting the total number of dots in the diagram for $n=4$, work out the fourth hexagonal number, ${H}_{4}$.