You are given a geometric progression (bn) in which b5 = 15, b8 = -1875. Find the denominator of the progression.

By condition:
b5 = 15;
b8 = – 1875.
We represent b5 as the product of the first term of the progression b1 and the denominator of the progression q:
b5 = b1 * q ^ 4.
We represent b8 as the product of the first term of the progression b1 and the denominator of the progression q:
b8 = b1 * q ^ 7.
Thus, one can come to a system of equations:
b1 * q ^ 4 = 15;
b1 * q ^ 7 = – 1875.
In the first equation of the system, we express b1 through q:
b1 = 15 / q ^ 4.
We substitute the resulting expression into the second equation of the system:
(15 / q ^ 4) * q ^ 7 = – 1875;
15q ^ 7 / q ^ 4 = – 1875;
15q ^ (7 – 4) = – 1875;
15q ^ 3 = – 1875;
q ^ 3 = – 1875/15;
q ^ 3 = – 125;
q = ³√ (- 125);
q = – 5.
Answer: q = – 5.