Lesson 12 1 Arithmetic Sequences A sequence is

Lesson 12. 1: Arithmetic Sequences • A sequence is an ordered set of numbers that has a one-to-one correspondence with the positive integers. • Each number in the sequence is called a term. • In general, a sequence can be written as where a 1 is the first term, a 2 is second term, an-1 is the second to last term, and an is the last term.

• The expression that describes the nth term, an , is often called an explicit or closed-form definition/rule for the sequence. Example: 2, 4, 6, 8, … can be written as an = 2 n

A sequence can also be defined as a function where the domain is restricted to the positive integers. In this case f(n) = an. The points (1, f(1)), (2, f(2)), (3, f(3)), … can be discretely graphed on the coordinate plane. Ex.

Recursively Defined Sequences • A recursively defined sequence will define an in terms of one or more previous terms. • To successfully use a recursive rule, one must know the first one or two terms used in the rule. Ex.

Arithmetic Sequences An arithmetic sequence is a sequence where consecutive terms have a common difference. The common difference is usually denoted d. Ex. 3, 10, 17, 24, 31, … Note: d = 7 Explicit Rule: an = 7 n – 4 Recursive Rule: a 1 = 3 and an = an-1 + 7

Writing Explicit Rules for Arithmetic Sequences Given an arithmetic sequence with first term a 1 and common difference d, one can write an explicit rule for the sequence as Ex. 3, 10, 17, 24, 31, … Since a 1 = 3 and d = 7, an = 3 + (n – 1)7 which can be simplified into an = 7 n – 4

Your Turn Write both an explicit rule and a recursive rule for the following sequence and then find the 13 th term. 35, 26, 17, 8, …