The ratio of the sixth term of a geometric progression to the second is 625. Find the denominator of this progression.

The denominator of the geometric progression is determined by the formula:
q = bn + 1 / bn;
Formula for the nth term of the geometric progression:
bn = b1 q ^ (n – 1);
The ratio is set:
b6 / b2 = 625;
b6 = b1 q ^ (6 – 1) = b1 q ^ 5;
b2 = b1 q ^ (2 – 1) = b1 q;
Let’s substitute these values into the original ratio:
b1 q ^ 5 / (b1 q) = q ^ 4 = 625;
q1 = 5.

q2 = -5;

Checking at q2 = -5.
b2 = b1 (-5) = -5 b1;
b6 = b1 (-5) ^ 5 = – 3125 b1;
b6 / b2 = – 3125 b1 / (- 5 b1) = 625;
The condition is fulfilled.
For q = 5 – this will be an increasing geometric progression, for q = -5 – alternating signs.
Answer: q = 5 or q = -5.