Glossary

13.3 Geometric Sequences

3 min read•june 24, 2024

Geometric sequences are mathematical patterns where each term is found by multiplying the previous term by a fixed number called the . These sequences are powerful tools for modeling growth or decay in various real-world scenarios.

Understanding geometric sequences opens doors to exponential functions and calculations. By mastering the , term generation, and formulas, you'll gain insights into patterns that appear in finance, biology, and physics.

Geometric Sequences

Common ratio in geometric sequences

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  • Sequence of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio (rr)
  • Find the common ratio by dividing any term in the sequence by the previous term
    • Formula: r=an+1[an](https://www.fiveableKeyTerm:an)r = \frac{a_{n+1}}{[a_n](https://www.fiveableKeyTerm:a_n)}, where ana_n is the nnth term of the sequence
  • Common ratio can be positive (2, 4, 8), negative (-3, 6, -12), or a fraction (1, 12\frac{1}{2}, 14\frac{1}{4})
    • If ∣r∣>1|r| > 1, terms increase in as nn increases (2, 6, 18)
    • If ∣r∣<1|r| < 1, terms decrease in as nn increases (80, 40, 20)

Term generation for geometric sequences

  • Generate terms by starting with the first term a1a_1 and multiplying by the common ratio rr for each subsequent term
    • a2=a1⋅ra_2 = a_1 \cdot r
    • a3=a2⋅r=a1⋅r2a_3 = a_2 \cdot r = a_1 \cdot r^2
    • a4=a3⋅r=a1⋅r3a_4 = a_3 \cdot r = a_1 \cdot r^3
  • General formula for the nnth term: an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}
  • Examples:
    • Sequence: 3, 6, 12, 24; a1=3a_1 = 3, r=2r = 2
    • Sequence: 1000, 100, 10, 1; a1=1000a_1 = 1000, r=110r = \frac{1}{10}

Recursive formulas for sequence analysis

  • defines each term in relation to the previous term
    • Formula: an=an−1⋅ra_n = a_{n-1} \cdot r, where a1a_1 is given
  • Extend a sequence using a recursive formula by multiplying the previous term by the common ratio to find the next term
    • Example: a1=5a_1 = 5, r=3r = 3; a2=5⋅3=15a_2 = 5 \cdot 3 = 15, a3=15⋅3=45a_3 = 15 \cdot 3 = 45
  • Useful for generating terms step-by-step and analyzing patterns in the sequence

Explicit formulas for specific terms

  • allows finding the nnth term directly without calculating previous terms
    • Formula: an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}, where a1a_1 is the first term, rr is the common ratio, and nn is the position of the term
  • Find a specific term by substituting values for a1a_1, rr, and nn into the formula and simplify
    • Example: a1=2a_1 = 2, r=4r = 4, find a5a_5
      1. a5=2⋅45−1a_5 = 2 \cdot 4^{5-1}
      2. a5=2⋅44a_5 = 2 \cdot 4^4
      3. a5=2⋅256=512a_5 = 2 \cdot 256 = 512
  • Useful for finding terms far into the sequence without calculating each preceding term (a50a_{50}, a100a_{100})

Connections to Exponential Functions and Applications

  • Geometric sequences are closely related to exponential functions, where the common ratio serves as the
  • The for a (an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}) is a discrete version of an
  • Exponents in geometric sequences represent repeated multiplication by the common ratio
  • Logarithms can be used to solve equations involving geometric sequences
  • Real-world applications include modeling in financial calculations

Key Terms to Review (37)

A_n: The term a_n, also known as the nth term, is a fundamental concept in mathematics that appears in various contexts, including power functions, polynomial functions, arithmetic sequences, and geometric sequences. It represents the value of a particular term or element within a sequence or function, where the subscript 'n' denotes the position or index of that term within the sequence.
Absolute value: Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is always non-negative and is denoted by vertical bars, e.g., $|x|$.
Absolute Value: Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of the number's sign. It is a fundamental operation that is essential in understanding and working with real numbers, radicals, and sequences.
Arithmetic-Geometric Sequence: An arithmetic-geometric sequence is a sequence that combines the properties of both an arithmetic sequence and a geometric sequence. It exhibits a linear pattern in the differences between consecutive terms, as well as an exponential pattern in the ratios between consecutive terms.
Base: The base is a fundamental component in various mathematical concepts, serving as a reference point or starting value. It is a crucial element in understanding exponents, exponential functions, logarithmic functions, and geometric sequences, among other topics.
Change-of-base formula: The change-of-base formula is used to rewrite logarithms in terms of logs of another base, allowing for easier computation. It is commonly written as $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ where $b$ and $c$ are positive real numbers and $c \neq 1$.
Common logarithm: A common logarithm is a logarithm with base 10, often written as $\log_{10}(x)$ or simply $\log(x)$. It is commonly used in scientific calculations and when dealing with exponential growth or decay.
Common ratio: The common ratio is the constant factor between consecutive terms of a geometric sequence. It is found by dividing any term by its preceding term.
Common Ratio: The common ratio is a constant value that represents the ratio between consecutive terms in a geometric sequence. It is the multiplicative factor that is used to generate each successive term in the sequence from the previous term.
Compound interest: Compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods. It is commonly used in finance and investments to calculate growth over time.
Compound Interest: Compound interest is the interest earned on interest, where the interest accrued on a principal amount is added to the original amount, and the total then earns interest in the next period. This concept is fundamental to understanding the growth of investments and loans over time.
Convergent Sequence: A convergent sequence is a sequence of numbers that approaches a specific value, known as the limit, as the terms of the sequence continue to increase or decrease. This concept is particularly important in the study of geometric sequences, where the ratio between consecutive terms determines the behavior of the sequence.
Divergent Sequence: A divergent sequence is a mathematical sequence where the terms move further and further apart, increasing without bound. This is in contrast to a convergent sequence, where the terms get closer and closer together, approaching a specific value.
Explicit formula: An explicit formula directly defines the nth term of a sequence as a function of n. Unlike recursive formulas, it does not require the computation of previous terms.
Explicit Formula: An explicit formula is a mathematical expression that directly defines a term or element in a sequence based on its position or index within the sequence. It provides a straightforward way to calculate any specific term without needing to refer to previous terms in the sequence.
Exponent: An exponent indicates how many times a number, known as the base, is multiplied by itself. It is written as a small number to the upper right of the base.
Exponent: An exponent is a mathematical symbol that indicates the number of times a base number is multiplied by itself. It represents the power to which a number or variable is raised, and it is a fundamental concept in algebra, exponential functions, logarithmic functions, and other areas of mathematics.
Exponential Decay: Exponential decay is a mathematical model that describes the gradual reduction or diminishment of a quantity over time. It is characterized by an initial value that decreases by a constant proportion during each successive time interval, resulting in an exponential decrease. This concept is fundamental to understanding various phenomena in fields such as physics, chemistry, biology, and finance.
Exponential function: An exponential function is a mathematical expression in the form $f(x) = a \cdot b^x$, where $a$ is a constant, $b$ is the base greater than 0 and not equal to 1, and $x$ is the exponent. These functions model growth or decay processes.
Exponential Function: An exponential function is a mathematical function in which the independent variable appears as an exponent. These functions exhibit a characteristic curve that grows or decays at a rate proportional to the current value, leading to rapid changes in output as the input increases.
Exponential growth: Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. This results in the function increasing rapidly over time.
Exponential Growth: Exponential growth is a pattern of change where a quantity increases at a rate proportional to its current value. This means the quantity grows by a consistent percentage over equal intervals of time, leading to rapid, accelerating growth. Exponential growth is a fundamental concept in mathematics and has applications across various fields, including biology, economics, and technology.
Finite Geometric Sequence: A finite geometric sequence is a sequence where each term is obtained by multiplying the previous term by a common ratio. The sequence has a finite number of terms and is used to model situations with exponential growth or decay.
Geometric Mean: The geometric mean is a type of average that is calculated by multiplying a set of numbers and then taking the nth root of the product, where n is the number of values in the set. It is particularly useful for measuring the central tendency of data that follows a log-normal distribution, such as in the context of geometric sequences.
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. The general form of a geometric sequence can be written as $a, ar, ar^2, ar^3, \ldots$.
Geometric Sequence: A geometric sequence is a sequence where each term is a constant multiple of the previous term. The ratio between consecutive terms is a fixed number, known as the common ratio, which determines the pattern of the sequence.
Geometric Series: A geometric series is a type of infinite series where each term is a constant multiple of the previous term. It is a sequence of numbers where the ratio between consecutive terms is constant, and it can be used to model various real-world phenomena that exhibit exponential growth or decay.
Infinite Geometric Sequence: An infinite geometric sequence is a sequence where each term is obtained by multiplying the previous term by a common ratio. This sequence continues indefinitely, with no end point.
Infinite Sum: An infinite sum, also known as a series, is the sum of an infinite sequence of numbers or terms. It represents the sum of a potentially endless series of addends, where each term is determined by a specific mathematical rule or pattern.
Logarithm: A logarithm is a mathematical function that describes the power to which a base number must be raised to get a certain value. It represents the exponent to which a base number must be raised to produce a given number. Logarithms are closely related to exponential functions and are essential in understanding topics such as logarithmic functions, graphs of logarithmic functions, exponential and logarithmic equations, and geometric sequences.
Nth partial sum: The nth partial sum of a series is the sum of the first n terms of that series. It provides an approximation to the total sum of an infinite series by summing a finite number of terms.
Nth Term: The nth term, also known as the general term, refers to the term in a sequence that corresponds to the nth position or index. It is a formula or expression that allows you to calculate any specific term in the sequence based on its position or index number.
Nth term of the sequence: The nth term of a sequence is a formula that allows you to find the value of any term in the sequence based on its position number, n. It generalizes the pattern within the sequence for all terms.
Partial Sum: The partial sum of a sequence or series is the sum of the first n terms of the sequence or series. It represents the accumulation of the terms up to a certain point, providing a snapshot of the overall sum as the sequence or series progresses.
R^n: The term $r^n$ represents the exponential expression where $r$ is the base and $n$ is the exponent. This expression is fundamental in the context of geometric sequences, as it describes the relationship between consecutive terms in the sequence.
Recursive Formula: A recursive formula is a mathematical expression that defines a sequence or series by relating each term to the previous term(s) in the sequence. It allows for the generation of a sequence by repeatedly applying the same rule or formula to generate the next term based on the preceding term(s).
Sum Formula: The sum formula is a mathematical expression used to calculate the sum of a sequence, particularly in the context of arithmetic and geometric sequences. It provides a concise way to determine the total value of a series of numbers without having to add them up individually.
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