You are given a geometric progression, the first term of which is 108. The sum of the first three members

You are given a geometric progression, the first term of which is 108. The sum of the first three members of this progression is 156. Find the 4th term of this progression.

From the formula for the sum of the first n terms of the progression, we will define its denominator:

Sn = b1 * (q ^ n – 1) / (q – 1).

S3 = b1 * (q ^ 3 – 1) / (q – 1) = (q – 1) * (q ^ 2 + q + 1) / (q – 1) = b1 * (q ^ 2 + q + 1 ) ..

156 = b1 * (q ^ 2 + q + 1) = 108 * (q ^ 2 + q + 1).

108 * q ^ 2 + 108 * q + 108 – 156 = 0.

108 * q ^ 2 + 108 * q– 48 = 0.

9 * q ^ 2 + 8 * q– 4 = 0.

Let’s solve the quadratic equation.

q1 = -4/3, then b4 = b1 * q ^ 3 = 108 * (-64/27) = -256.

q2 = 1/3, then b4 = b1 * q ^ 3 = 108 / (-1/27) = -4.

Answer: The fourth term is -256 or -4.