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8.1: Sequences

Learn about sequences in mathematics, including explicit and recursive definitions, arithmetic and geometric sequences, and how to plot sequences on a Ti-89 calculator. Includes examples and explanations.

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8.1: Sequences

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  1. Photo by Vickie Kelly, 2008 Greg Kelly, Hanford High School, Richland, Washington 8.1: Sequences Craters of the Moon National Park, Idaho

  2. A sequence is a list of numbers written in an explicit order. nth term Any real-valued function with domain a subset of the positive integers is a sequence. If the domain is finite, then the sequence is a finite sequence. In calculus, we will mostly be concerned with infinite sequences.

  3. Example: A sequence is defined explicitly if there is a formula that allows you to find individual terms independently. To find the 100th term, plug 100 in for n:

  4. Example: A sequence is defined recursively if there is a formula that relates an to previous terms. We find each term by looking at the term or terms before it: You have to keep going this way until you get the term you need.

  5. An arithmetic sequence has a common difference between terms. Example: Arithmetic sequences can be defined recursively: or explicitly:

  6. An geometric sequence has a common ratio between terms. Example: Geometric sequences can be defined recursively: or explicitly:

  7. If the second term of a geometric sequence is 6 and the fifth term is -48, find an explicit rule for the nth term. Example:

  8. ENTER Sequence Graphing on the Ti-89 Change the graphing mode to “sequence”: MODE Graph……. 4

  9. Y= alpha Use the key to enter the letter n. Example: Plot Leave ui1 blank for explicitly defined functions.

  10. WINDOW

  11. WINDOW GRAPH

  12. Y= alpha Use the key to enter the letters u and n. The previous example was explicitly defined. Now we will use a recursive definition to plot the Fibonacci sequence. Enter the initial values separated by a comma (even though the comma doesn’t show on the screen!)

  13. WINDOW Enter the initial values separated by a comma (even though the comma doesn’t show on the screen!)

  14. WINDOW GRAPH You can use F3 Trace to investigate values.

  15. TBLSET TABLE We can also look at the results in a table. Scroll down to see more values.

  16. TABLE Scroll down to see more values.

  17. Does converge? You can determine if a sequence converges by finding the limit as n approaches infinity. The sequence converges and its limit is 2.

  18. Absolute Value Theorem for Sequences If the absolute values of the terms of a sequence converge to zero, then the sequence converges to zero. Don’t forget to change back to function mode when you are done plotting sequences. p

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