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 \( a_1 \) of the geometric sequence. In this sequence, the first term is \( 2 \).
2. Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{6}{2} \).
3. Recall the formula for the sum of the first \( n \) terms of a geometric sequence: \[ S_n = a_1 \times \frac{1 - r^n}{1 - r} \] where \( n \) is the number of terms.
4. Substitute the values \( a_1 = 2 \), \( r \) (from step 2), and \( n = 12 \) into the formula to set up the expression for \( S_{12} \).
5. Simplify the expression by calculating \( r^{12} \), then compute the numerator and denominator separately before dividing to find the sum of the first 12 terms.

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, in the sequence 2, 6, 18, 54, the common ratio is 3 because each term is three times the previous one.

Sum of the First n Terms of a Geometric Sequence

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 a is the first term, r is the common ratio, and n is the number of terms. This formula helps find the total sum without adding each term individually.

Exponents and Powers

Exponents represent repeated multiplication of a base number. In the sum formula, r^n means multiplying the common ratio r by itself n times. Understanding how to work with exponents is essential for correctly applying the sum formula and simplifying expressions.