Today in Precalculus Notes: Sequences Homework Go over quiz

Vocabulary and notation Sequences: an ordered progression of numbers. Term: each number in a sequence is a term First term is a 1 Second term is a 2 n th term is a n The subscripts denote only the position of the term in the sequence.

Types Arithmetic Sequence: a sequence in which there is a common difference, d, between every pair of successive terms. Example: 5,8,11,14 Geometric: a sequence in which there is a common ratio, r, between every pair of successive terms. Example:

Types Infinite: there is an infinite number of terms in the sequence Example: Finite: a finite number of terms in the sequence. Example: 5,8,11,14 Sequences are infinite unless otherwise specified.

Explicitly Defined Sequence A formula is given for any term in the sequence Example: a k = 2k - 5 Find the first 5 terms and the 20 th term for the sequence a 1 = 2(1) – 5 = – 3 a 2 = 2(2) – 5 = – 1 a 3 = 2(3) – 5 = 1 a 4 = 2(4) – 5 = 3 a 5 = 2(5) – 5 = 5 a 20 = 2(20) – 5 = 35

Recursively Defined Sequence The first term is given and along with a rule to obtain each succeeding term from the one preceding it. Example: b 1 = 8 and b n = b n-1 – 2 for all n>1 Find the next 4 terms for the sequence b 2 = b 1 – 2 = 8 – 2 = 6 b 3 = 6 – 2 = 4 b 4 = 4 – 2 = 2 b 5 = 2 – 2 = 0

General formulas for finding terms in a sequence Arithmetic: a n = a 1 + (n – 1)d Geometric: a n = a 1 r (n–1) To use these: 1) Determine if the sequence is arithmetic or geometric 2) Find the common difference or ratio

Example 1 Find the 20 th term of the sequence 55,49,43, … and write a recursive and explicit rule. Arithmetic sequence with d= -6 a 20 = 55 + (20 – 1)(-6) a 20 = –59 Recursive rule: a k = a k-1 – 6 Explicit rule: a n = 55 + (n – 1)(-6) a n = 55 – 6n + 6 a n = 61 – 6n

Example 2 Find the 8 th term of the sequence and write a recursive and explicit rule. Geometric sequence with r=4 Recursive rule: a k = 4a k-1 Explicit rule:

Homework Pg 739: 1-9odd, 21-31odd