The difference between the second and third terms of the geometric progression is 18,

The difference between the second and third terms of the geometric progression is 18, and their sum is 54. Determine the first term and denominator of the progression.

Let’s use the formula for the nth term of the geometric progress: bn = b1 * q ^ n-1
then: b2 = b1 * q and b3 = b1 * q ^ 2
We get a system of two equations:
b1 * q ^ 2-b1 * q = 18
b1 * q ^ 2 + b1 * q = 54
Let us express b1 from the 1st equation:
b1 (q ^ 2-q) = 18
b1 = 18 / (q ^ 2-q)
and substitute in the 2nd:
(q ^ 2 + q) / (q ^ 2-q) = 54/18
q ^ 2 + q = 3 * (q ^ 2-q)
2q ^ 2-2q = 0
q-1 = 0
q = 1
Then 1 * b1 + 1 * b1 = 54
2 * b1 = 54
b1 = 27