Exponentially (An) A5 = 1, A7 = 0.25. Find the denominator.

Let’s use the formula for the nth term of the geometric progression bn = b1 * qn – 1, where b1 is the first term of the geometric progression, q is the denominator of the geometric progression.

According to the condition of the problem, in this geometric progression, the fifth term is a5 = 1, and the seventh term is a7 = 0.25.

Applying the n-th term of the geometric progression in the formula for n = 5 and n = 7, we obtain the following relations:

b1 * q ^ 5 – 1 = 1;

b1 * q ^ 7 – 1 = 0.25.

Dividing the second ratio by the first, we get:

b1 * q ^ 6 / (b1 * q ^ 4) = 0.25 / 1;

q ^ 6 / q ^ 4 = 0.25;

q ^ 2 = 0.25;

q ^ 2 = (0.5) ^ 2.

This quadratic equation has two solutions: q = -0.5 and q = 0.5.

Answer: The denominator of this geometric progression can take two values: -0.5 and 0.5.