Mathematics
Grade 9
15 min
Write variable expressions for arithmetic sequences
Write variable expressions for arithmetic sequences
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Introduction & Learning Objectives
Learning Objectives
Define an arithmetic sequence and identify its key components.
Identify the first term (a₁) and the common difference (d) from a given sequence.
Write an explicit formula (a variable expression) for an arithmetic sequence using the formula aₙ = a₁ + (n - 1)d.
Simplify the explicit formula into a linear expression.
Use the variable expression to find the value of any term in the sequence.
Model a real-world scenario with an arithmetic sequence and write a variable expression to represent it.
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This tutorial will teach you how to describe number patterns called arithmetic sequences wi...
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Key Concepts & Vocabulary
TermDefinitionExample
Arithmetic SequenceAn ordered list of numbers in which the difference between any two consecutive terms is constant.The sequence 4, 9, 14, 19, 24, ... is an arithmetic sequence because you add 5 to get to the next term every time.
TermEach individual number in a sequence. We use subscript notation like aₙ to denote the term in the nth position.In the sequence 4, 9, 14, 19, ..., the first term is a₁ = 4, the second term is a₂ = 9, and so on.
Common Difference (d)The constant value that is added to each term to get the next term in an arithmetic sequence. It can be positive or negative.In the sequence 4, 9, 14, 19, ..., the common difference is d = 5. In the sequence 20, 17, 14, ..., the common difference is d = -3.
First Term (a₁)The starting value of a sequence.In th...
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Core Formulas
Explicit Formula for an Arithmetic Sequence
aₙ = a₁ + (n - 1)d
This is the primary formula used to write a variable expression for an arithmetic sequence. 'aₙ' is the value of the term you want to find, 'a₁' is the first term, 'n' is the position of the term, and 'd' is the common difference.
Finding the Common Difference
d = a₂ - a₁
To find the common difference, subtract any term from the term that immediately follows it. The simplest way is to subtract the first term from the second term.
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Challenging
Sequence A is defined by aₙ = 3n + 5. Sequence B is defined by bₙ = 5n - 3. For what value of n is the nth term of Sequence A equal to the nth term of Sequence B?
A.n = 2
B.n = 3
C.n = 4
D.n = 5
Challenging
An arithmetic sequence is represented by aₙ = 4n - 11. Which statement about the terms of this sequence is true?
A.All terms are positive.
B.All terms are odd.
C.All terms are even.
D.The sequence contains both positive and negative terms.
Challenging
A freelance designer charges a $100 setup fee plus $45 for each hour of work. If 'h' represents the number of hours, which expression correctly models the total cost for 'h' hours, framed as an arithmetic sequence where a_h is the cost?
A.a_h = 45h + 55
B.a_h = 100h + 45
C.a_h = 45h + 100
D.a_h = 145h
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