1. Introduction
Arithmetic sequences are linear functions that have a domain of positive consecutive integers in which the difference between any two consecutive terms is equal. Arithmetic sequences can be represented by formulas, either explicit or recursive, and those formulas can be used to find a certain term of the sequence or the number of a certain value in the sequence. An explicit formula is a formula used to find the nth term of a sequence and a recursive formula is a formula used to find the next term of a sequence when the previous term is known.
3.8.1: Arithmetic Sequences

2. Key Concepts
An arithmetic sequence is a list of terms separated by a common difference, the number added to each consecutive term in an arithmetic sequence.
An arithmetic sequence is a linear function with a domain of positive consecutive integers in which the difference between any two consecutive terms is equal.
The rule for an arithmetic sequence can be expressed either explicitly or recursively.
3.8.1: Arithmetic Sequences

3. Key Concepts, continued
The explicit rule for an arithmetic sequence is an = a1 + (n – 1)d, where a1 is the first term in the sequence, n is the term, d is the common difference, and an is the nth term in the sequence.
The recursive rule for an arithmetic sequence is an = an – 1 + d, where an is the nth term in the sequence, an – 1 is the previous term, and d is the common difference.
3.8.1: Arithmetic Sequences

4. Common Errors/Misconceptions
identifying a non–arithmetic sequence as arithmetic
defining the common difference, d, in a decreasing sequence as a positive number
incorrectly using the distributive property when finding the nth term with the explicit formula
forgetting to identify the first term when defining an arithmetic sequence recursively
3.8.1: Arithmetic Sequences

5. Guided Practice Example 2
Write a linear function that corresponds to the following arithmetic sequence.
8, 1, –6, –13, …
3.8.1: Arithmetic Sequences

6. Guided Practice: Example 2, continued
Find the common difference by subtracting two successive terms.
1 – 8 = –7
3.8.1: Arithmetic Sequences

7. Guided Practice: Example 2, continued
Confirm that the difference is the same between all of the terms.
–6 – 1 = –7 and –13 – (–6) = –7
3.8.1: Arithmetic Sequences

8. Guided Practice: Example 2, continued Identify the first term (a1).
3.8.1: Arithmetic Sequences

9. Guided Practice: Example 2, continued Write the explicit formula.
an = a1 + (n – 1)d Explicit formula for any given arithmetic sequence
an = 8 + (n – 1)(–7) Substitute values for a1 and d.
3.8.1: Arithmetic Sequences

10. Guided Practice: Example 2, continued Simplify the explicit formula.
an = 8 – 7n Distribute –7 over (n – 1).
an = –7n Combine like terms.
3.8.1: Arithmetic Sequences

11. ✔ Guided Practice: Example 2, continued
Write the formula in function notation.
ƒ(x) = –7x + 15
Note that the domain of an arithmetic sequence is positive consecutive integers. ✔
3.8.1: Arithmetic Sequences

12. Guided Practice: Example 2, continued 12
3.8.1: Arithmetic Sequences

13. Guided Practice Example 3
An arithmetic sequence is defined recursively by an = an – 1 + 5, with a1 = 29. Find the first 5 terms of the sequence, write an explicit formula to represent the sequence, and find the 15th term.
3.8.1: Arithmetic Sequences

14. Guided Practice: Example 3, continued Using the recursive formula:
a2 = a1 + 5
a2 = = 34
a3 = = 39
a4 = = 44
a5 = = 49
The first five terms of the sequence are 29, 34, 39, 44, and 49.
3.8.1: Arithmetic Sequences

15. Guided Practice: Example 3, continued
The first term is a1 = 29 and the common difference is d = 5, so the explicit formula is an = 29 + (n – 1)5.
3.8.1: Arithmetic Sequences

16. Guided Practice: Example 3, continued Simplify.
an = n – 5
an = 5n + 24    Combine like terms.
3.8.1: Arithmetic Sequences

17. ✔ Guided Practice: Example 3, continued
Substitute 15 in for n to find the 15th term in the sequence.
a15 = 5(15) + 24
a15 =
a15 = 99
The 15th term in the sequence is 99. ✔
3.8.1: Arithmetic Sequences

18. Guided Practice: Example 3, continued
3.8.1: Arithmetic Sequences