1. The hexagonal numbers are a sequence of numbers. The first six hexagonal numbers are

${H}_{1}=1,{H}_{2}=6,{H}_{3}=15,{H}_{4}=28,{H}_{5}=45,{H}_{6}=66$

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

$\frac{1+\sqrt{8\times(number)+1}}{4}$

If the output is a whole number, then it is a hexagonal number.

For example, to test if 6 is a hexagonal number, substitute it into the formula:

$\begin{matrix}\frac{1+\sqrt{8\times6+1}}{4} & = & \frac{1+\sqrt{48+1}}{4}\\ & = & \frac{1+\sqrt{49}}{4}\\ & = & \frac{1+7}{4}\\ & = & \frac{8}{4}\\ & = & 2\end{matrix}$

The output is 2, which is a whole number, so 6 is a hexagonal number.

Use the test above to determine whether 120 is a hexagonal number.

You must show your working clearly.