ࡱ> egd AbjbjΚΚ rA  TDlc>sss,S!sQ"ssss^s"J`S=:30cBM"M"JJ|sssssscM"sssssssss  :   Pre - Calculus 11 ( Ch 1: Sequences and Series Name:_________________  Many patterns and designs linked to mathematics are found in nature and the human body. Certain patterns occur more often than others. Logistic spirals, such as the Golden Mean spiral, are based on the Fibonacci number sequence. The Fibonacci sequence is often called Natures Numbers. The pattern of this logistic spiral is found in the chambered nautilus, the inner ear, star clusters, cloud patterns, and whirlpools. Seed growth, leaves on stems, petals on flowers, branch formations, and rabbit reproduction also appear to be modeled after this logistic spiral pattern. There are many different kinds of sequences. In this chapter, you will learn about sequences that can be described by mathematical rules. The Fibonacci sequence is a sequence of natural numbers named after Leonardo of Pisa, also known as Fibonacci. Each number is the sum of the two preceding numbers. 1, 1, 2, 3, 5, 8, 13, . . . Lesson Notes 1.1: Arithmetic Sequences Objectives: deriving a rule for determining the general term of an arithmetic sequence determining t1, d, n, or tn in a problem that involves an arithmetic sequence describing the relationship between an arithmetic sequence and a linear function solving a problem that involves an arithmetic sequence What is a sequence? A sequence is an ordered list of numbers. Each number is referred as an element or term that follows a pattern or rule to determine the next term in the sequence. The terms of a sequence are labeled according to their position in the sequence. For example: i) 7, 10, 13, 16, 19, ____, ____, ____. Rule: ____________________________________ ii) 2, 5, 12, 19, _____, _____, ______. Rule: ____________________________________ iii) 5, 10, 15, 20, . . . . . . . . . . . . . . . Rule: ____________________________________ A _______________________ always has a finite number of terms, like: __________. An ______________________ has an infinite number of terms, just like: __________. The three examples shown from above are typical examples of Arithmetic Sequences. An arithmetic sequence is an ordered list of terms in which the difference between consecutive terms is constant. In other words, there is a common difference between two consecutive terms. Look at the following examples of Non- Arithmetic Sequences. a) 2, 4, 6, 10, 16, _____, ______, _______. Rule:_______________________________ b) 3, 4, 6, 7, 9, _____, ______, ______. Rule: ______________________________ Consider the sequence 10, 16, 22, 28, . . . . Terms t1 t2 t3  t4  Sequence 10 16  22 28 Sequence Expressed Using First Term and Common Difference  10 10 + (6) 10 + (6) + (6) 10 + (6) + (6) + (6) General Sequence t1 t2 =  t3 =  t4 = The formula for the general term helps you find the terms of a sequence. This formula is a rule that shows how the value of tn depends on n. The general arithmetic sequence is t1, t1 + d, t1 + 2d, t1 + 3d, . . ., where t1 is the first term and d is the common difference. t1 = t1 t2 = t1 + d t3 = t1 + 2d . . tn = t1 + (n 1)d The general term of an arithmetic sequence is tn = t1 + (n 1)d where t1 is the first term of the sequence n is the number of terms d is the common difference tn is the general term or nth term  Example 1) Write the first four terms of each arithmetic sequence for the given values of t1 and d. a) t1 = 4, d = 5 b) t1 = 2, d = 3 Example 2) Identify the arithmetic sequences from the following sequences. For each arithmetic sequence, state the value of t1, the value of d, and the next three terms. a) 16, 32, 48, 64, 80, . . . b) 2, 4, 8, 16, 32, . . . Example 3) Insert 3 numbers between -5 and 11 to form an arithmetic sequence. Example 4) A visual and performing arts group wants to hire a community events leader. The person will be paid $12 for the first hour of work, $19 for two hours of work, $26 for three hours of work, and so on. a) Write the general term that you could use to determine the pay for any number of hours worked. b) What will the person get paid for 6 h of work? Example 5) Many factors affect the growth of a child. Medical and health officials encourage parents to keep track of their childs growth. The general guideline for the growth in height of a child between the ages of 3 years and 10 years is an average increase of 5 cm per year. Suppose a child was 70 cm tall at age 3. a) Write the general term that you could use to estimate what the childs height will be at any age between 3 and 10. b) How tall is the child expected to be at age 10? 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