Question

Prove that in any square pyramid, the squares of the height, slant height and lateral edge are in arithmetic sequence.

Answer 1

If we look into a square pyramid, we can identify the following two right-angled triangles:

where $b$, $e$, $h$ and $l$ represent the lengths of the base edge, lateral edge, height and the slant height respectively.
Using Pythagoras theorem, we have the following two relations:
$l^2+\frac{1}{2}{b}^2=e^2$ and
$h^2+\frac{1}{2}{b}^2=l^2$$\left[1mark\right]$
In other words, $e^2-l^2=l^2-h^2=\frac{1}{2}{b}^2$.
Thus, the squares of the height, slant height and lateral edge are in arithmetic sequence with common difference $\frac{1}{2}{b}^2$.$\left[1mark\right]$