Find the sum of the first 15 terms of the geometric sequence: 5, -15, 45, -135

Verified step by step guidance

1. Identify the first term \( a_1 \) of the geometric sequence. Here, \( a_1 = 5 \).
2. Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{-15}{5} \).
3. Use 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 \( n = 15 \).
4. Substitute the values \( a_1 = 5 \), \( r \) from step 2, and \( n = 15 \) into the sum formula.
5. Simplify the expression by calculating \( r^{15} \), then perform the subtraction and division inside the formula to find the sum \( S_{15} \).

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 5, -15, 45, -135, the common ratio is -3 because each term is multiplied by -3 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. Knowing the common ratio is essential for finding terms and sums in the sequence.

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 efficiently without adding each term individually.