1.1. The sequence of hexagonal numbers, ${H}_{n}$, is shown below.

A new sequence, ${K}_{n}$, is formed using the $n$th term formula:

${K}_{n}=n+{H}_{n}$

The first three terms, ${K}_{1}$, ${K}_{2}$ and ${K}_{3}$, are shown in the table below.

Find ${K}_{4}$, ${K}_{5}$ and ${K}_{6}$.

1.2. A fraction is said to be a perfect square if it can be written in the form ${(\frac{a}{b})}^{2}$ where $a$ and $b$ are non-zero integers.

Terms in the sequence of ${K}_{n}$ can be made into fractions by dividing the previous term, ${K}_{n-1}$, by the current term:

$\frac{{K}_{n-1}}{{K}_{n}}=\frac{previousterm}{currentterm}$

Use the table below to show that fractions formed in this way are perfect squares.

The first example has been done for you, as follows:

$\frac{{K}_{1}}{{K}_{2}}=\frac{2}{8}=\frac{1}{4}={(\frac{1}{2})}^{2}$

1.3. By substituting the $n$th term formula

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

into the formula

${K}_{n}=n+{H}_{n}$

and simplifying, use algebra to prove that

$\frac{{K}_{n-1}}{{K}_{n}}=\frac{{(n-1)}^{2}}{{n}^{2}}$