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

July 28, 2021 | education

| **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.

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