Three positive numbers form a geometric progression. If its first term is multiplied by -3, then the resulting

Three positive numbers form a geometric progression. If its first term is multiplied by -3, then the resulting numbers will form an arithmetic progression in the same order. What is the denominator of the original geometric progression?

Let us express the first number, for which we use the variable a.

Therefore, according to the condition of the task, we can write the second number as aq (where q is the sought denominator), and the third number can be represented as aq ^ 2.

Then the elements of the arithmetic progression will look like this:

-3a, aq, aq ^ 2.

Using the property of the arithmetic progression, we write the equation:

aq – (-3a) = aq ^ 2 – aq;

q + 3 = q ^ 2 – q;

q ^ 2 – 2q – 3 = 0;

D = 4 – 4 * (-3) = 16 = 4 ^ 2;

q1 = (2 + 4): 2 = 3;

q2 = (2 – 4): 2 = -1 (does not match by condition).

Answer: 3.