Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30.Find the sum of the first 12 terms of the geometric sequence: 2, 6, 18, 54, ...

Verified step by step guidance

1. Identify the first term \( a \) of the geometric sequence, which is 2.
2. Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{6}{2} = 3 \).
3. Use the formula for the sum of the first \( n \) terms of a geometric sequence: \( S_n = a \frac{1-r^n}{1-r} \).
4. Substitute the known values into the formula: \( S_{12} = 2 \frac{1-3^{12}}{1-3} \).
5. Simplify the expression to find the sum of the first 12 terms.

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the given sequence, the common ratio can be determined by dividing any term by its preceding term, which in this case is 3 (6/2, 18/6, etc.). Understanding this concept is crucial for identifying the pattern and calculating the sum of the terms.

Sum of the First n Terms

The sum of the first n terms of a geometric sequence can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where S_n is the sum, a is the first term, r is the common ratio, and n is the number of terms. This formula allows for efficient calculation of the sum without needing to add each term individually, making it essential for solving problems involving geometric sequences.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term. It is a key component in both identifying the sequence and applying the sum formula. In the provided sequence, the common ratio is 3, which is critical for accurately calculating the sum of the first 12 terms using the appropriate formula.