The third term of the geometric progression is 4 times less than its first term.

The third term of the geometric progression is 4 times less than its first term. How many times is the ninth term of this progression less than its third term?

Let’s denote the number that is in the first position in this sequence through b1, and the denominator of this geometric progression through q.

In the initial data for this task, it is reported that the number that is located in the given sequence at the third position is four times less than b1, therefore, the following relationship holds:

b1 * q ^ 3 / b1 = 4,

whence follows:

q ^ 3 = 4.

Let’s find the ratio of the number that is located in the given sequence at the ninth position to the number that is located in the given sequence at the third position:

b1 * q ^ 8 / (b1 * q ^ 2) = q ^ 8 / q ^ 2 = q ^ 6 = (q3) ^ 2 = 4 ^ 2 = 16.

Answer: 16 times.