### 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.

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.

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.