### Theory:

Summing up two consecutive natural numbers results in a square:

Consider any odd natural number, say 5. Its square is ${5}^{2}$ $$=$$ 25

To find two consecutive natural numbers which is equal to the odd square, we can use the below formula.

${a}^{2}\phantom{\rule{0.147em}{0ex}}=\frac{{a}^{2}-1}{2}+\phantom{\rule{0.147em}{0ex}}\frac{{a}^{2}+1}{2}$

Now let's take $$a =$$ 5.

Substitute the $$a$$ value in the formula.

$\begin{array}{l}{5}^{2}\phantom{\rule{0.147em}{0ex}}=\frac{{5}^{2}-1}{2}+\phantom{\rule{0.147em}{0ex}}\frac{{5}^{2}+1}{2}\\ \\ 25=\frac{24}{2}+\frac{26}{2}\\ \\ 25=12\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}13\\ \\ 25=25\end{array}$

Therefore the two consecutive natural numbers are 13 and 12, which is equal to the square of odd number 25.

Let's see another example.

The odd square number is 17. Square of 17 is 289.

Therefore the we can substitute $$a =$$17 in the formula.

$\begin{array}{l}{17}^{2}\phantom{\rule{0.147em}{0ex}}=\frac{{17}^{2}-1}{2}+\phantom{\rule{0.147em}{0ex}}\frac{{17}^{2}+1}{2}\\ \\ 289=\frac{288}{2}+\frac{290}{2}\\ \\ 289=144+145\\ \\ 289=289\end{array}$

Important!
Note: If the value of $$a$$ is an even number, then the two natural numbers will be decimals.
From the above examples, we can see that each of their squares can be written as the sum of two consecutive natural numbers.
"The square of an odd number can be written as the sum of two consecutive natural numbers".