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Sequences & Summation Notation 8.1

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  1. Sequences & Summation Notation8.1 JMerrill, 2007 Revised 2008

  2. Sequences In Elementary School… 12 12 32

  3. And… 17 12

  4. Even 12 22

  5. Sequences • SEQUENCE - a set of numbers, called terms, arranged in a particular order.


  6. 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.

  7. Examples • Finite sequence: 2, 6, 10, 14 • Infinite sequence:

  8. 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

  9. 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

  10. You Do • Write the first 4 terms of the sequence

  11. 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

  12. 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

  13. 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.

  14. 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.

  15. 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

  16. 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

  17. Example • Write the first 4 terms of the sequence • a0 = 1 • a1 = 2 • a2 = 2 • a3 = 4/3 • a4 = 2/3

  18. 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

  19. Example • Write the first four terms of the sequence

  20. Evaluating Factorials in Fractions • Evaluate:

  21. 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

  22. Examples • Finite sequence: 2, 6, 10, 14 • Finite series: 2 + 6 + 10 + 14 • Infinite sequence: • Infinite series:

  23. 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.”

  24. 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

  25. 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.

  26. Definitions Limits of Summation Summand Index of Summation

  27. Example

  28. Sigma Notation Representing Infinite Series

  29. Give the series in expanded form: 5+10+15+20

  30. Find the Sum of 190 Calculator steps: in LIST

  31. One More: Find the Sum of 1089

  32. Properties of Sums

  33. Last Problem • Find the sum of