Triangular Numbers

By Chandrahas M. Halai

This is in continuation of my earlier article:

In the earlier article I had described the Triangular Numbers. The 5th triangular number is given by:

This is shown in Figure 1.

Figure 1

The spheres can be rearranged as shown in Figure 2.

Figure 2

Now, the 6th triangular number is

This is as shown in figure 3.

Figure 3

Now, let us add the 5th and 6th triangular numbers, we get:

This is as shown in figure 4.

Figure 4

Let us now add the 7th and 8th triangular numbers:

Now let us add the 10th and 11th triangular numbers:

From the above discussion we can say that adding two consecutive triangular numbers gives a square number. That is

Now, let us prove this.

We have

Hence, we have