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## Sequences & Summation Notation 8.1

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**Sequences & Summation Notation8.1**JMerrill, 2007 Revised 2008**Sequences In Elementary School…**12 12 32**And…**17 12**Even**12 22**Sequences**• SEQUENCE - a set of numbers, called terms, arranged in a particular order.**Sequences**• An infinite sequence is a function whose domain is the set of positive integers. The function values a1, a2, a3, …, an… are the terms of the sequence. • If the domain of the sequence consists of the first n positive integers only, the sequence is a finite sequence. n is the term number.**Examples**• Finite sequence: 2, 6, 10, 14 • Infinite sequence:**Writing the Terms of a Sequence**• Write the first 4 terms of the sequence an = 3n – 2 • a1 = 3(1) – 2 = 1 • a2 = 3(2) – 2 = 4 • a3 = 3(3) – 2 = 7 • a4 = 3(4) – 2 = 10 Calculator steps in LIST**Writing the Terms of a Sequence**• Write the first 4 terms of the sequence an = 3 + (-1)n • a1 = 3 + (-1)1 =2 • a2 = 3 + (-1)2 =4 • a3 = 3 + (-1)3 =2 • a4 = 3 + (-1)4 =4**You Do**• Write the first 4 terms of the sequence**Consider the infinite sequence**Because a sequence is a function whose domain is the set of positive integers, the graph of a sequence is a set of distinct points. The first term is ½ , the 2nd term is ¼ … So, the ordered pairs are (1, ½ ), (2, ¼ )… Graphs**Finding the nth Term of a Sequence**• Write an expression for the nth term (an) of the sequence 1, 3, 5, 7… • n: 1, 2, 3, 4…n • Terms: 1, 3, 5, 7…an • Apparent pattern: each term is 1 less than twice n. So, the apparent nth term is • an = 2n - 1 Always compare the term to the term number**Finding the nth Term of a SequenceYou Do**• Write an expression for the nth term (an) of the sequence • Apparent pattern: n = 1, 2, 3, 4…n The numerator is 1; the denominator is the square of n.**Recursive Definition**• Sometimes a sequence is defined by giving the value of an in terms of the preceding term, an-1. A recursive sequence consists of 2 parts: • An initial condition that tells where the sequence starts. • A recursive equation (or formula) that tells how many terms in the sequence are related to the preceding term.**Example**• If an = an-1 + 4 and a1 = 3, give the first five terms of the sequence. • a1 = 3 • If n = 2: a2= a1+ 4 = 3 + 4 = 7 • If n = 3: a3 = a2+ 4 = 7 + 4 = 11 • If n = 4: a4= a3+ 4 = 11 + 4 = 15 • If n = 5: a5= a4 + 4 = 15 + 4 = 19**A Famous Recursive Sequence**• The Fibonacci Sequence is very well known because it appears in nature. • The sequence is 1, 1, 2, 3, 5, 8, 13… • Apparent pattern? • Each term is the sum of the preceding 2 terms • The nth term is • an = an-2 + an-1**Example**• Write the first 4 terms of the sequence • a0 = 1 • a1 = 2 • a2 = 2 • a3 = 4/3 • a4 = 2/3**Factorial Notation**• Products of consecutive positive integers occur quite often in sequences. These products can be expressed in factorial notation: • 1! = 1 • 2! = 2 ● 1 = 2 • 3! = 3 ●2 ●1 = 6 • 4! = 4 ●3 ●2 ●1 = 24 • 5! = 5 ●4 ●3 ●2 ●1 = 120 The factorial key can be found in MATH PRB:4 on your calculator 0!, by definition, = 1**Example**• Write the first four terms of the sequence**Evaluating Factorials in Fractions**• Evaluate:**Definitions**• The words sequences and series are often used interchangeably in everyday conversation. (A person may refer to a sequence of events or a series of events.) In mathematics, they are very different. • Sequence: a set of numbers, terms, arranged in a particular order • Series: the sum of a sequence**Examples**• Finite sequence: 2, 6, 10, 14 • Finite series: 2 + 6 + 10 + 14 • Infinite sequence: • Infinite series:**Intro to Sigma**• The Greek letter (sigma) is often used in mathematics to represent a sum (series) in abbreviated form. • Example: which can be read as “the sum of k2 for values of k from 1 to 100.”**Definition of a Series**• Consider the infinite series a1, a2, … an… • The sum of the first n terms is a finite series (or partial sum) and is denoted by • The sum of all terms of an infinite sequence is called an infinite series and is denoted by**Sigma Continued**• Similarly, the symbol is read “the sum of 3k for values of k from 5 to 10.” This means that the symbol represents the series whose terms are obtained by evaluating 3k for k = 5, k = 6, and so on, to k = 10.**Definitions**Limits of Summation Summand Index of Summation**Give the series in expanded form:**5+10+15+20**Find the Sum of**190 Calculator steps: in LIST**Last Problem**• Find the sum of