number 36 appears in both lists. But there is a more striking link. If you take
successive triangular numbers and add them together, what do you get? Let’s try
it and put the results in a table.
Adding two successive triangular numbersThat’s right! When you add two successive triangular numbers together you
get a square number. You can also see this with a ‘proof without words’.
Consider a square made up of 4 rows of 4 dots with a diagonal line drawn
through it. The dots above the line (shown) form a triangular number and below
the line is the next triangular number. This observation holds for any sized
square. It’s a short step from these ‘dotty diagrams’ to measuring areas. The area
of a square whose side is 4 is 4 × 4 = 4^2 = 16 square units. In general, if the
side is called x then the area will be x^2.
The square x^2 is the basis for the parabolic shape. This is the shape you find
in satellite receiver dishes or the reflector mirrors of car headlights. A parabola
has a focus point. In a receiving dish a sensor placed at the focus receives the