Section 5.7 Arithmetic and Geometric Sequences

1. Section 5.7Arithmetic and Geometric Sequences

2. What You Will Learn • Arithmetic Sequences • Geometric Sequences

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

4. Arithmetic Sequence • An arithmetic sequenceis 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 sequenceis 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 a1and common ratio r, the general or nth term can be found using the following formula. • an = a1rn–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 = a1rn–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 = a1rn–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