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13.2 Arithmetic Sequences

2 min read•june 24, 2024

Arithmetic sequences are lists of numbers with a between terms. They're like staircases, where each step is the same height. You can find any by knowing the first term and the .

These sequences show up in many real-life situations, like savings accounts or population growth. You can use formulas to find specific terms or generate the whole , making them super useful in math and everyday life.

Arithmetic Sequences

Common difference in arithmetic sequences

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  • Value added to each term to obtain the next term in the sequence
  • Calculated by subtracting any term from the following term d=[an+1](https://www.fiveableKeyTerm:an+1)−[an](https://www.fiveableKeyTerm:an)d = [a_{n+1}](https://www.fiveableKeyTerm:a_{n+1}) - [a_n](https://www.fiveableKeyTerm:a_n)
  • Can be positive (increasing sequence), negative (decreasing sequence), or zero (constant sequence)
  • Determines the rate of change between
  • Remains constant throughout the entire sequence (linear growth or decay)

Generation of arithmetic sequence terms

  • Defined by the first term [a1](https://www.fiveableKeyTerm:a1)[a_1](https://www.fiveableKeyTerm:a_1) and dd
  • Generate terms by starting with a1a_1 and repeatedly adding dd
    • a2=a1+da_2 = a_1 + d
    • a3=a1+2da_3 = a_1 + 2d
    • a4=a1+3da_4 = a_1 + 3d
  • Terms increase by a fixed amount (common difference) at each step
  • Can be used to model situations with constant rates of change (population growth, savings accounts)
  • Each element in the sequence is called a term

Recursive formulas for arithmetic sequences

  • Define each term based on the previous term and common difference
  • : an+1=an+da_{n+1} = a_n + d
    • Next term obtained by adding dd to the current term
  • Requires knowledge of the first term a1a_1 and common difference dd
  • Allows for step-by-step generation of sequence terms
  • Example: a1=3a_1 = 3, d=2d = 2, recursive formula: an+1=an+2a_{n+1} = a_n + 2
    1. a2=a1+2=3+2=5a_2 = a_1 + 2 = 3 + 2 = 5
    2. a3=a2+2=5+2=7a_3 = a_2 + 2 = 5 + 2 = 7
    3. a4=a3+2=7+2=9a_4 = a_3 + 2 = 7 + 2 = 9

Explicit formulas for sequence terms

  • Find the nn-th term directly without calculating previous terms
  • : an=a1+(n−1)da_n = a_1 + (n - 1)d
    • ana_n: nn-th term
    • a1a_1: first term
    • nn: position of the term
    • dd: common difference
  • Substitute values for a1a_1, nn, and dd to find a specific term
  • Useful for finding terms far along in the sequence without iterating through all previous terms
  • Example: a1=3a_1 = 3, d=2d = 2, find the 10th term (n=10n = 10)
    • a10=3+(10−1)2a_{10} = 3 + (10 - 1)2
    • a10=3+18=21a_{10} = 3 + 18 = 21

Understanding Sequences and Series

  • A sequence is an ordered list of numbers following a specific
  • An is a type of where the difference between consecutive terms is constant
  • The sum of terms in a sequence forms a
  • Arithmetic sequences can be used to model various real-world patterns

Key Terms to Review (25)

A_{n+1}: In the context of arithmetic sequences, $a_{n+1}$ represents the next term in the sequence, which is calculated by adding the common difference to the previous term. This term is crucial for understanding the pattern and predicting future terms in an arithmetic sequence.
A_1: In the context of arithmetic sequences, $a_1$ represents the first term or the starting value of the sequence. It is the initial value from which the sequence is generated by adding a constant difference, known as the common difference, to each successive term.
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.
Arithmetic Mean: The arithmetic mean, commonly referred to as the average, is a measure of central tendency that represents the sum of all the values in a dataset divided by the total number of values. It is a widely used statistical concept that provides a single value to summarize and describe the central tendency of a group of numbers.
Arithmetic sequence: An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference.
Arithmetic Sequence: An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference, and it allows the sequence to be generated by adding the common difference to each term.
Common difference: The common difference is the constant amount added or subtracted between consecutive terms in an arithmetic sequence. It is denoted by $d$.
Common Difference: The common difference, denoted by the letter 'd', is the constant value that is added or subtracted between consecutive terms in an arithmetic sequence. It represents the consistent change or rate of change between each term in the sequence.
Consecutive Terms: Consecutive terms refer to a sequence of numbers or values where each term is directly following the previous term, with a constant difference between any two adjacent terms. This concept is particularly important in the context of arithmetic sequences, where the difference between consecutive terms is a fixed value known as the common difference.
Constant Difference: The constant difference is a key characteristic of an arithmetic sequence, where the difference between consecutive terms remains the same throughout the sequence. This consistent difference is a defining feature that allows for the predictable pattern and calculation of the sequence.
Decreasing linear function: A decreasing linear function is a linear function where the value of the function decreases as the input increases. It has a negative slope.
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.
Finite Arithmetic Sequence: A finite arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant, and the sequence has a finite number of terms. This type of sequence is commonly studied in the context of 13.2 Arithmetic Sequences, which explores the properties and applications of arithmetic sequences.
Geometric series: A geometric series is the sum of the terms of a geometric sequence, where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Infinite Arithmetic Sequence: An infinite arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This sequence continues indefinitely, with no end point.
Linear Function: A linear function is a mathematical function that represents a straight line on a coordinate plane. It is characterized by a constant rate of change, known as the slope, and can be expressed in the form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.
Nth Term Formula: The nth term formula is a mathematical expression used to represent the general term or pattern in an arithmetic sequence. It allows for the prediction of any term in the sequence based on its position or index within the sequence.
Pattern: A pattern is a repeated or predictable arrangement of numbers or elements that can be identified within a sequence. In the context of arithmetic sequences, patterns help us understand how each term relates to others and how to derive formulas for finding terms in the sequence.
Progression: In mathematics, progression refers to a sequence of numbers arranged in a specific order according to a particular rule. One key type of progression is an arithmetic sequence, where each term after the first is found by adding a constant difference to the previous term. This consistent pattern makes it easy to identify and calculate terms within 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).
Sequence: A sequence is an ordered list of elements, such as numbers, letters, or objects, that follow a specific pattern or rule. Sequences are fundamental concepts in mathematics and are extensively studied in various topics, including algebra, calculus, and discrete mathematics.
Series: A series is a sequence of numbers or terms that follow a specific pattern or rule. It is a way of representing a sum of infinitely many terms, where each term is related to the previous one through a defined relationship.
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.
Term: A term is a distinct component or part of a mathematical expression, such as a number, variable, or operation. In the context of arithmetic sequences, a term refers to each individual element or number within the sequence.
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13.3 Geometric Sequences