A finite geometric series is a type of series where the ratio between consecutive terms is constant. It is a sum of a finite number of terms in a geometric sequence, where each term is obtained by multiplying the previous term by a common ratio.
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The formula for the sum of a finite geometric series is $S_n = a(1 - r^n) / (1 - r)$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
Finite geometric series converge (have a finite sum) when the common ratio $|r| < 1$, and diverge (have an infinite sum) when $|r| \geq 1$.
Finite geometric series are often used to model real-world phenomena, such as compound interest, depreciation, and population growth.
The sum of a finite geometric series can be used to find the total value of a series of payments or the total amount of an investment over time.
Finite geometric series are an important concept in calculus and are often used in the study of series and sequences.
Review Questions
Explain the relationship between a finite geometric series and a geometric sequence.
A finite geometric series is the sum of a finite number of terms in a geometric sequence. The terms in a geometric sequence are generated by multiplying the previous term by a constant ratio, called the common ratio. Similarly, the terms in a finite geometric series are obtained by adding these geometric sequence terms, where the common ratio between consecutive terms is also constant. The formula for the sum of a finite geometric series is derived from the properties of a geometric sequence.
Describe the conditions under which a finite geometric series will converge or diverge.
A finite geometric series will converge (have a finite sum) when the common ratio $|r| < 1$. In this case, as the number of terms increases, the contribution of each successive term becomes smaller, and the sum approaches a finite value. Conversely, a finite geometric series will diverge (have an infinite sum) when the common ratio $|r| \geq 1$. This is because the terms in the series either remain constant or increase in magnitude, causing the sum to grow without bound as the number of terms increases.
Analyze how finite geometric series can be used to model real-world phenomena and solve practical problems.
Finite geometric series are widely used to model various real-world situations, such as compound interest, depreciation, and population growth. For example, the formula for the sum of a finite geometric series can be used to calculate the total value of a series of payments or the total amount of an investment over time, taking into account the constant growth or decay rate. Similarly, finite geometric series can be used to model the growth or decline of a population, where the common ratio represents the rate of change. Understanding the properties and applications of finite geometric series is crucial for solving a variety of practical problems in fields like finance, economics, and biology.
Related terms
Geometric Sequence: A sequence where the ratio between consecutive terms is constant.
Common Ratio: The constant ratio between consecutive terms in a geometric sequence.
Infinite Geometric Series: A geometric series that continues indefinitely, where the number of terms approaches infinity.