Theory:

Like how we have dealt with arithmetic progression, we will deal with geometric progression in this section.
Situation 1
In this situation, the midpoints of \(\triangle ABC\) is joined to form \(\triangle DEF\).
 
Similarly, the midpoint of \(\triangle CEF\) is joined to form \(\triangle MNO\).
 
The midpoints of \(\triangle CON\) is joined to form a different triangle and so on.
 
1.svg
 
The area of \(\triangle ABC\), \(\triangle DEF\), \(\triangle MNO\) and so on is given by \(\triangle ABC\), \(\frac{1}{4}\triangle ABC\), \(\frac{1}{4} \times \frac{1}{4} \triangle ABC\) and so on.
 
That is, \(\triangle ABC\), \(\frac{1}{4} \triangle ABC\), \(\frac{1}{16} \triangle ABC\), and so on.
 
In other words, the areas of \(\triangle ABC\), \(\triangle DEF\), \(\triangle MNO\) and so on are \(\frac{1}{4}\) apart.
 
Therefore, the areas of the triangles formed are in a geometric progression with \(\frac{1}{4}\) as the common ratio.
Situation 2
A particular dog breed gives birth to exactly two puppies at a time.
 
The number of dogs in each of the levels is given as \(1\), \(2\), \(4\),\(…\)
 
Therefore, the number of dogs is a geometric progression with \(2\) as the common ratio.
 
2.svg
Geometric progression
A Geometric Progression is a sequence in which each term is obtained by multiplying a fixed non-zero number to the preceding term except the first term. The fixed number is called common ratio. The common ratio is usually denoted by \(r\).
The general form of geometric progression:
 
A geometric progression is given in the form of \(a\), \(ar\), \(ar^2\),\(...ar^{n-1}\).
 
Here, \(a\) is the first term, and \(r\) is the common ratio.
 
The first term \(a\), when multiplied by the \(r\) subsequently, forms \(a\), \(ar\), \(ar^2\),\(...ar^{n-1}\), which is the geometric progression.