1. Section 5.7 Arithmetic and Geometric Sequences

2. What You Will Learn
Arithmetic Sequences
Geometric Sequences

3. Sequences
A sequence is a list of numbers that are related to each other by a rule.
The terms are the numbers that form the sequence.

4. Arithmetic Sequence
An arithmetic sequence is a sequence in which each term after the first term differs from the preceding term by a constant amount.
The common difference, d, is the amount by which each pair of successive terms differs.
To find the difference, simply subtract any term from the term that directly follows it.

5. Example 2: An Arithmetic Sequence with a Negative Difference
Write the first five terms of the arithmetic sequence with first term 9 and a common difference of –4.
Solution
The first five terms of the sequence are
9, 5, 1, –3, –7

6. General or nth Term of an Arithmetic Sequence
For an arithmetic sequence with first term a1 and common difference d, the general or nth term can be found using the following formula.
an = a1 + (n – 1)d

7. Example 3: Determining the 12th Term of an Arithmetic Sequence
Determine the twelfth term of the arithmetic sequence whose first term is –5 and whose common difference is 3.
Solution
Replace: a1 = –5, n = 12, d = 3
an = a1 + (n – 1)d
a12 = –5 + (12 – 1)3
= –5 + (11)3
= 28

8. Example 4: Determining an Expression for the nth Term
Write an expression for the general or nth term, an, for the sequence 1, 6, 11, 16,…
Solution
Substitute: a1 = 1, d = 5
an = a1 + (n – 1)d
= 1 + (n – 1)5
= 1 + 5n – 5
= 5n – 4

9. Sum of the First n Terms of an Arithmetic Sequence
The sum of the first n terms of an arithmetic sequence can be found with the following formula where a1 represents the first term and an represents the nth term.

10. Example 5: Determining the Sum of an Arithmetic Sequence
Determine the sum of the first 25 even natural numbers.
Solution
The sequence is 2, 4, 6, 8, 10, …, 50
Substitute a1 = 2, a25 = 50, n = 25 into the formula

11. Example 5: Determining the Sum of an Arithmetic Sequence
Solution
a1 = 2, a25 = 50, n = 25

12. Geometric Sequences
A geometric sequence is one in which the ratio of any term to the term that directly precedes it is a constant.
This constant is called the common ratio, r.
r can be found by taking any term except the first and dividing it by the preceding term.

13. Example 6: The First Five Terms of a Geometric Sequence
Write the first five terms of the geometric sequence whose first term, a1, is 5 and whose common ratio, r, is 2.
Solution
The first five terms of the sequence are
5, 10, 20, 40, 80

14. General or nth Term of a Geometric Sequence
For a geometric sequence with first term a1 and common ratio r, the general or nth term can be found using the following formula.
an = a1r n–1

15. Example 7: Determining the 12th Term of a Geometric Sequence
Determine the twelfth term of the geometric sequence whose first term is –4 and whose common ratio is 2.
Solution
Replace: a1 = –4, n = 12, r = 2
an = a1r n–1
a12 = –4 • 212–1
= –4 • 211 = –4 • 2048
= –8192

16. Example 8: Determining an Expression for the nth Term
Write an expression for the general or nth term, an, for the sequence 2, 6, 18, 54,…
Solution
Substitute: a1 = 2, r = 3
an = a1r n–1
= 2(3)n–1

17. Sum of the First n Terms of an Geometric Sequence
The sum of the first n terms of an geometric sequence can be found with the following formula where a1 represents the first term and r represents the common ratio.

18. Example 9: Determining the Sum of an Geometric Sequence
Determine the sum of the first five terms in the geometric sequence whose first term is 4 and whose common ratio is 2.
Solution
Substitute a1 = 4, r = 2, n = 5 into

19. Example 5: Determining the Sum of an Arithmetic Sequence
Solution
a1 = 2, r = 2, n = 5