Section 5.7 Arithmetic and Geometric Sequences

What You Will Learn Arithmetic Sequences Geometric Sequences

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.

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.

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

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

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

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

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.

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

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

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.

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

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

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

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

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.

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

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