Let's solve the given problem step by step.

Problem Analysis:

We are tasked with finding the common ratio of a geometric progression (геометрична прогресия) where:

1. $a_n > 0$, so all terms are positive.
2. It is given that: $a_5 - a_1 = 5 \cdot (a_3 - a_1)$

Key formula for a geometric progression:

The general term of a geometric progression can be expressed as: $a_n = a_1 \cdot q^{n-1}$ where $q$ is the common ratio.

Express the terms:

• $a_1 = a_1$
• $a_3 = a_1 \cdot q^{2}$
• $a_5 = a_1 \cdot q^{4}$

Now substitute these into the equation $a_5 - a_1 = 5 \cdot (a_3 - a_1)$: $a_1 \cdot q^4 - a_1 = 5 \cdot \left( a_1 \cdot q^2 - a_1 \right)$

Step 1: Factorize $a_1$:

$a_1 \cdot (q^4 - 1) = 5 \cdot a_1 \cdot (q^2 - 1)$

Since $a_1 > 0$, we can divide through by $a_1$: $q^4 - 1 = 5 \cdot (q^2 - 1)$

Step 2: Simplify:

Expand both sides: $q^4 - 1 = 5q^2 - 5$

Rearrange to form a standard equation: $q^4 - 5q^2 + 4 = 0$

Step 3: Substitute $x = q^2$:

Let $x = q^2$, so the equation becomes: $x^2 - 5x + 4 = 0$

Factorize: $(x - 4)(x - 1) = 0$

Thus: $x = 4 \quad \text{or} \quad x = 1$

Step 4: Solve for $q$:

Since $x = q^2$, we have: $q^2 = 4 \quad \Rightarrow \quad q = 2 \quad \text{or} \quad q = -2$ $q^2 = 1 \quad \Rightarrow \quad q = 1 \quad \text{or} \quad q = -1$

However, since $q > 0$ (given that $a_n > 0$), the only valid solutions are: $q = 2 \quad \text{or} \quad q = 1$

Final Step:

For a geometric progression, $q = 1$ would make all terms the same, which contradicts the setup of the problem. Hence: $q = 2$

Final Answer:

The common ratio $q$ is: $\boxed{2 \, (\text{Answer B})}$

Let me know if you'd like detailed clarifications!

Related Questions:

1. How can we derive the general formula for a geometric progression?
2. What happens if $q$ is negative in a geometric sequence?
3. Can a geometric progression have $q = 1$? Why or why not?
4. How do we verify if a sequence is geometric?
5. What are some real-world examples of geometric progressions?

Tip:

Always simplify equations step by step, especially when working with sequences or progressions!