You are given a geometric progression (Bn). Calculate the sum of the first 3 terms if b4 = 54, q = 3.

Given a geometric progression {bn}, for which the equalities b4 = 54 and q = 3. As required by the task, calculate the sum of the first three terms of this geometric progression.
As you know, in order to be able to have a complete picture of the geometric progression {bn}, it is enough to know only two of its parameters: the first term b1 and the denominator q. In the task, one of these parameters is given, namely, q = 3. Calculate b1, using the formula bn = b1 * qn – 1. We have: b4 = b1 * q4 – 1 or 54 = b1 * 3 ^ 3, whence b1 = 54/27 = 2.
In order to calculate the required sum of the first three terms of this geometric progression, we use the following formula: Sn = b1 * (1 – qn) / (1 – q), where q ≠ 1. We have: S3 = b1 * (1 – q ^ 3) / (1 – q) = 2 * (1 – 3³) / (1 – 3) = 2 * (1 – 27) / (-2) = 26.
It should be noted that the required sum can be calculated by simple summation of b1, b2 and b3, for which, first, using the definition of a geometric progression, we calculate b2 = b1 * q = 2 * 3 = 6 and b3 = b2 * q = 6 * 3 = 18 . Then, S3 = b1 + b2 + b3 = 2 + 6 + 18 = 26.
Answer: 26.