1 / 42

Sequences and Series (Section 9.4 in Textbook)

Sequences and Series (Section 9.4 in Textbook). Arithmetic Sequences. How are these sets related?. 3, 6, 9, 12, 15, 18, 21, 24, … 25, 20, 15, 10, 5, 0, -5, -10, … -21, 3, 27, 51, 75, 99, 123, …. Arithmetic Sequence.

clarencet
Télécharger la présentation

Sequences and Series (Section 9.4 in Textbook)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Sequences and Series(Section 9.4 in Textbook)

  2. Arithmetic Sequences

  3. How are these sets related? • 3, 6, 9, 12, 15, 18, 21, 24, … • 25, 20, 15, 10, 5, 0, -5, -10, … • -21, 3, 27, 51, 75, 99, 123, …

  4. Arithmetic Sequence An arithmetic sequence is defined as a sequence in which there is a common difference between consecutive terms. Recursive Formula: an = an-1 + d Common Difference = 5

  5. Is the given sequence arithmetic? If so, identify the common difference. • 2, 4, 8, 16, … • 4, 6, 12, 18, 24, … • 2, 5, 7, 12, … • 48, 45, 42, 39, … • 1, 4, 9, 16, … • 10, 20, 30, 40, …

  6. Arithmetic Sequence Explicit Formula The 1st number in the sequence. The common difference. an = a1 + (n – 1) • d The same as the n in an. If you’re looking for the 5th number in the sequence, n = 5. The “nth” number in the sequence. Ex. a5 is the 5th number in the sequence.

  7. Examples: an = a1 + (n – 1) • d 1) Given the sequence -4, 5, 14, 23, 32, 41, 50,…, find the 14th term. 2) Given the sequence 79, 75, 71, 67, 63,…, find the term number that is -169.

  8. For the arithmetic sequence: -5, -2, 1, 4,… find: • The common difference: • The 10th term: • Recursive rule for the nth term: • Explicit rule for the nth term:

  9. Example: an = a1 + (n – 1) • d Suppose you are saving up for a new gaming system. You have 100 dollars this year, and you plan to add 33 dollars each of the following years. How much money will you have in 7 years?

  10. Constructing Sequences The 4th and 7th terms of an arithmetic sequence are -8 and 4, respectively. Find the 1st term and a explicit rule for the nth term.

  11. Geometric Sequences

  12. Geometric sequences are different. See if you can spot the relationship! • 3, 6, 12, 24, 48, 96,… • 81, 27, 9, 3, 1, ⅓,… • -2, 4, -8, 16, -32, 64, -128

  13. Geometric Sequences An geometric sequence is defined as a sequence in which there is a common ratio between consecutive terms. • an = an-1r Recursive Formula: Common Ratio = 2

  14. Geometric Sequence Formula The 1st number in the sequence. The same as the n in an. If you’re looking for the 5th number in the sequence, n = 5. an = a1 • r (n-1) The “nth” number in the sequence. Ex. a5 is the 5th number in the sequence. The common ratio.

  15. Examples: an = a1 • r (n-1) 1) Given the sequence 4, 28, 196, 1372, 9604,…, find the 14th term. 2) Given the sequence 1, 5, 25, 125, 625,…, find the term number that is 9,765,625.

  16. Example : an = a1 • r (n-1) Suppose you want a reduced copy of a photograph. The actual length of the photograph is 10 in. The smallest size the copier can make is 64% of the original. Find the length of the photograph after five reductions.

  17. You Try! For the geometric sequence: 1, -2, 4, -8, 16,… find: • The common ratio: • The 8th term: • Recursive rule for the nth term: • Explicit rule for the nth term:

  18. Fibonacci Sequence The Fibonacci sequence can be defined recursively by: a1 = 1; a2 = 1; an = an-2 + an-1(For all positive integers n ≥ 3) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Link: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

  19. Sum of numbers 1-100?

  20. Series and Summation Notation

  21. Series A series is the sum of the terms in a sequence. • 3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 = 135 • 3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 = 135

  22. Series Formulas • Geometric Series • ArithmeticSeries a1•(1 – rn) (1 – r) n 2 ________ Sn = – (a1 + an) Sn =

  23. Evaluate a series with the terms 1, 7, 13, 19, 25 for the first 13 terms. • Find the sum of the first 10 terms of the geometric series with a1 = 6 and r = 2.

  24. A philanthropist donates $50 to the SPCA. Each year, he pledges to donate 12 dollars more than the previous year. In 8 years, what is the total amount he will have donated?

  25. Summation Notation Instead of saying: “Find the sum of the series denoted by an = 3n + 2 from the 3rd term to 7th term,” mathematicians made up a symbol to deal with it. Sigma! ∑ I’m just a fancy way of saying, “Add everything up!”

  26. “Find the sum of the series denoted by • an = 3n + 2 from the 3rd term to 7th term” • now looks like: last term sequence formula first term

  27. Evaluating Using Summation Notation

  28. Ex) Use summation notation to write each series for the specified number of terms. 3 + 8 + 13 + 18 + …; n = 9

  29. Sum of a Finite Arithmetic Sequence Let {a1, a2, a3, ….} be a finite arithmetic sequence with common difference d. Then the sum of the terms of the sequence is

  30. Find the sum of the arithmetic sequence: 1, 2, 3, 4, ……, 80

  31. Rewrite the sum using Sigma Notation 5 + 9 + 13 +17 + … + 85

  32. Sum of a Finite Geometric Sequence Let {a1, a2, a3, ….., an} be a finite geometric sequence with common ratio r ≠ 1. Then the sum of the terms of the sequence is

  33. Ex) Find the sum of the geometric sequence 3, 6, 12, ……, 12288

  34. Infinite Geometric Series Geometric series are special. Sometimes we can find their sum, even if they go on forever. In order to do this, we need to decide if the series is convergent or divergent.

  35. Convergent Geometric Series The following series are convergent because the terms eventually approach 0.

  36. Divergent Geometric Series The following series are divergent because the terms do not have a limit.

  37. Sum of an Infinite Geometric Series The geometric series Converges if and only if . If it does converge, the sum is S=

  38. Examples Determine whether each infinite geometric series diverges or converges. If it converges, find the sum. a) 1 – 1/3 + 1/9 - … b) 4 + 8 + 16 + …

  39. Determine whether the infinite geometric series converges. If it does, find the sum.

  40. Rational Numbers Repeating decimals are considered rational numbers because they can be represented as a ratio of two integers. A number is rational if you can write it in a form a/b where a and b are integers, b not zero. 

  41. Since 0.11111... = 1/9, then the decimal number 0.11111... is a rational number.   In fact, every non-terminating decimal number that REPEATS a certain pattern of digits, is a rational number.  For example, let's make up a decimal number 0.135135135135135... that never ends.  Do you believe we CAN write it as a fraction, in the form a/b?

  42. Express the rational numbers as a fractions of integers 1) 7.14141414 2) -17.268268268

More Related