A geometric sequence is a sequence of numbers where each term is a constant multiple of the previous term. The ratio between consecutive terms remains the same throughout the sequence, creating a pattern of exponential growth or decay.
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The formula for the $n$th term of a geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.
The common ratio $r$ is the constant ratio between any two consecutive terms in the sequence, and it determines whether the sequence is increasing (|$r$| > 1) or decreasing (|$r$| < 1).
Geometric sequences can be used to model real-world situations such as population growth, radioactive decay, and compound interest.
The sum of a finite geometric sequence can be calculated using the formula $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$, where $S_n$ is the sum of the first $n$ terms.
Infinite geometric series can be used to represent recurring decimal expansions, and their sum can be calculated using the formula $S = \frac{a_1}{1 - r}$ if |$r$| < 1.
Review Questions
Explain the relationship between the common ratio and the behavior of a geometric sequence.
The common ratio $r$ is the key determinant of the behavior of a geometric sequence. If the common ratio |$r$| > 1, the sequence is increasing exponentially, while if |$r$| < 1, the sequence is decreasing exponentially. The magnitude of the common ratio also affects the rate of growth or decay in the sequence. For example, a sequence with $r = 2$ will grow much faster than a sequence with $r = 1.1$, even though both are increasing. Conversely, a sequence with $r = 0.5$ will decrease more rapidly than a sequence with $r = 0.9$. Understanding the role of the common ratio is crucial for analyzing and applying geometric sequences in real-world contexts.
Describe how the formulas for the $n$th term and the sum of a finite geometric sequence are derived and explain their practical applications.
The formula for the $n$th term of a geometric sequence, $a_n = a_1 \cdot r^{n-1}$, is derived by recognizing the pattern of multiplying the first term $a_1$ by the common ratio $r$ repeatedly. This allows us to determine the value of any term in the sequence given the first term and the common ratio. This formula has practical applications in modeling situations where the value of each term is a constant multiple of the previous term, such as population growth, radioactive decay, and compound interest. The formula for the sum of a finite geometric sequence, $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$, is derived by recognizing that the sum of the terms in a geometric sequence can be expressed as a geometric series. This formula allows us to efficiently calculate the total sum of a finite number of terms in a geometric sequence, which is useful in a variety of real-world applications involving accumulation or depletion over time.
Explain the significance of the infinite geometric series formula $S = \frac{a_1}{1 - r}$ and how it relates to the representation of recurring decimal expansions.
The formula for the sum of an infinite geometric series, $S = \frac{a_1}{1 - r}$, is particularly important because it allows us to calculate the sum of an infinite number of terms in a geometric sequence, provided that the common ratio |$r$| < 1. This formula has a direct connection to the representation of recurring decimal expansions. Any repeating decimal can be expressed as a fraction, and the denominator of that fraction can be written as a sum of powers of 10, which is a geometric series. By applying the infinite geometric series formula, we can then simplify the fraction and express the recurring decimal in a more compact form. This relationship between infinite geometric series and recurring decimals is a powerful tool in mathematics, allowing us to work with and understand these types of numerical representations more effectively.