Find the sum of the first 15 terms of the geometric sequence: 5, -15, 45, -135
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 first term is 5, and the common ratio can be determined by dividing any term by its preceding term, which in this case is -3.
The sum of the first n terms of a geometric series 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.
The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term. It is crucial for determining the behavior of the sequence and calculating the sum of its terms. In the sequence provided, the common ratio is -3, which indicates that the terms alternate in sign and increase in absolute value.