Mathematics
Grade 11
15 min
Write a formula for an arithmetic sequence
Write a formula for an arithmetic sequence
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Introduction & Learning Objectives
Learning Objectives
Define an arithmetic sequence and identify its first term (a₁) and common difference (d).
Derive and write the explicit formula for the nth term of an arithmetic sequence.
Write a formula for an arithmetic sequence when given the first term and the common difference.
Write a formula for an arithmetic sequence when given two non-consecutive terms.
Use the formula to find the value of any specific term in a sequence.
Model a real-world linear growth scenario by writing an arithmetic sequence formula.
Ever notice how a stack of concert tickets, numbered consecutively, follows a perfect pattern? 🎟️ That predictable pattern can be described with a simple formula!
In this tutorial, you will learn how to capture the pattern of any arithmetic sequence using a si...
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Key Concepts & Vocabulary
TermDefinitionExample
Arithmetic SequenceA sequence of numbers in which the difference between any two consecutive terms is constant.The sequence 3, 7, 11, 15, 19, ... is an arithmetic sequence because you add 4 to each term to get the next.
Term (aₙ)Each individual number in a sequence. The variable 'n' represents the term's position in the sequence.In the sequence 3, 7, 11, 15, ..., the third term is 11, which we denote as a₃ = 11.
First Term (a₁)The very first number in a sequence.In the sequence 3, 7, 11, 15, ..., the first term is a₁ = 3.
Common Difference (d)The constant value that is added to each term to get the next term. It can be positive, negative, or zero.In 3, 7, 11, 15, ..., the common difference is d = 4. In 50, 45, 40, 35, ..., the common difference is d =...
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Core Formulas
Explicit Formula for an Arithmetic Sequence
aₙ = a₁ + (n - 1)d
This is the fundamental formula used to find the value of the nth term (aₙ) of an arithmetic sequence. You need the first term (a₁), the common difference (d), and the position of the term you want to find (n).
Formula for the Common Difference
d = (aₘ - aₙ) / (m - n)
Use this formula to find the common difference (d) when you are given the values of two non-consecutive terms, aₘ and aₙ, at positions m and n.
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Challenging
The first three terms of an arithmetic sequence are x+1, 2x+3, and 5x-2. What is the explicit formula for this sequence?
A.aₙ = 5.5n + 1
B.aₙ = 5.5n - 1
C.aₙ = 4.5n + 1
D.aₙ = 4.5n - 1
Challenging
In an arithmetic sequence, the sum of the 3rd and 5th terms is 28, and the 9th term is 34. What is the explicit formula for the sequence?
A.aₙ = 4n - 2
B.aₙ = 3n + 7
C.aₙ = 4n + 2
D.aₙ = 5n - 11
Challenging
An arithmetic sequence is defined by aₙ = a₁ + (n-1)d. A new sequence, bₙ, is created by taking every third term of the original sequence, starting with a₂. (i.e., b₁=a₂, b₂=a₅, b₃=a₈, ...). What is the explicit formula for bₙ in terms of a₁ and d?
A.bₙ = (a₁ + d) + (n-1)d
B.bₙ = a₂ + (n-1)(2d)
C.bₙ = a₁ + (n-1)(3d)
D.bₙ = (a₁ + d) + (n-1)(3d)
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