What numbers are both square numbers and triangular numbers?

Answer 1

The only square number that is also a triangular number is 1. This is because square numbers are of the form n^2, while triangular numbers are of the form (n*(n+1))/2. When setting these two equations equal to each other and solving for n, we find that n=1 is the only integer solution. Therefore, 1 is the only number that is both a square number and a triangular number.

Answer 2

There are an infinite number of triangular numbers that are also squares. You can generate such numbers using the formula:

Sn+1 = 4Sn(8Sn + 1)

With the first Sn being 1.

This gives the next such number as:

Sn+1= 4 (8 + 1)

= 36

You can then plug that into the equation to get:

Sn+1 = 4 x 36 (8 x 36 + 1)

= 41616

and so on.

Answer 3

36