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Geometric Sequences

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Geometric sequences are numerical patterns where each term is derived by multiplying the previous one by a constant, known as the common ratio. This fundamental concept is crucial in fields like finance, biology, and physics, where it helps model exponential growth or decay. Understanding the common ratio allows for the prediction and calculation of sequence terms, making geometric sequences a key mathematical tool.

Understanding Geometric Sequences

A geometric sequence, or geometric progression, is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant called the common ratio. This ratio is non-zero and can be either positive or negative, leading to sequences that increase, decrease, oscillate, or even approach zero. Geometric sequences exemplify exponential growth or decay and are essential in various applications, such as computing compound interest, tracking population dynamics, and modeling half-life in radioactive decay. The behavior of a geometric sequence is largely determined by the common ratio, making it a pivotal component in understanding and utilizing these sequences.
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Characteristics of Geometric Sequences

In a geometric sequence, each term is called a member of the sequence, and the sequence is defined by its first term and the common ratio. The ratio between any two consecutive terms is constant, which is the hallmark of a geometric sequence. For example, in the sequence 2, 6, 18, 54, ..., each term is obtained by multiplying the previous term by 3. This constant factor, the common ratio, remains the same throughout the sequence, providing a predictable pattern that is useful in mathematical analysis and practical applications.

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00

The ______ ratio of a geometric sequence dictates whether the sequence will grow, shrink, oscillate, or approach ______.

common

zero

01

Definition of a geometric sequence member

A member in a geometric sequence is a term within the sequence.

02

First term significance in a geometric sequence

The first term is the starting point and helps determine all subsequent terms.

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