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 14 terms of the geometric sequence: - 3/2, 3, - 6, 12, ...

Verified step by step guidance

1. Identify the first term \( a_1 \) of the geometric sequence. Here, the first term is \( -\frac{3}{2} \).
2. Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{3}{-\frac{3}{2}} \).
3. Recall the formula for the sum of the first \( n \) terms of a geometric sequence: \[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \] where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
4. Substitute \( a_1 = -\frac{3}{2} \), \( r \) (from step 2), and \( n = 14 \) into the sum formula.
5. Simplify the expression step-by-step to find the sum of the first 14 terms, being careful with signs and powers of \( r \).

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 -3/2, 3, -6, 12, ..., each term is multiplied by -2 to get the next term.

Common Ratio

The common ratio in a geometric sequence is the fixed factor between consecutive terms. It is found by dividing any term by the previous term. Identifying the common ratio is essential for applying the sum formula and understanding the sequence's behavior.

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 of the sequence's terms efficiently.