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Section 9.1 – Sequences

Section 9.1 – Sequences. Sequence. A sequence is a list of numbers written in an explicit order. First Term. n th Term. Second Term. Generally, we will concentrate on infinite sequences , that is, sequences with domains that are infinite subsets of the positive integers.

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Section 9.1 – Sequences

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  1. Section 9.1 – Sequences

  2. Sequence A sequence is a list of numbers written in an explicit order. First Term nth Term Second Term Generally, we will concentrate on infinite sequences, that is, sequences with domains that are infinite subsets of the positive integers.

  3. Recursive Formula A formula that requires the previous term(s)in order to find the value of the next term. Example: Find a Recursive Formula for the sequence below. 2, 4, 8, 16, …

  4. Explicit Formula A formula that requires the number of the term in order to find the value of the next term. Example: Find an Explicit Formula for the sequence below. 2, 4, 8, 16, … The Explicit Formula is also known as the General or nth Term equation.

  5. Arithmetic Sequences A sequence which has a constant difference between terms. The rule is linear. Example: 1, 4, 7, 10, 13,… (generator is +3) Discrete +3 +3 +3 Explicit Formula Recursive Formula +3

  6. White Board Challenge Write an equation for the nth term of the sequence: 36, 32, 28, 24, … Then find the n=0 term. n=1 n=2 n=3 n=4 n=0 40, – 4 First find the generator Sequences typically start with n=1 a(0) is not in the sequence! Do not include it in tables or graphs!

  7. Geometric Sequences A sequence which has a constant ratio between terms. The rule is exponential. Example: 4, 8, 16, 32, 64, … (generator is x2) x2 Discrete x2 0 1 2 3 4 5 6 x2 Explicit Formula Recursive Formula x2

  8. White Board Challenge Write an equation for the nth term of the sequence: Then find the n=0 term. n=0 n=1 n=2 n=3 n=4 3, 15, 75, 375, … x5 First find the generator Sequences typically start with n=1 a(0) is not in the sequence! Do not include it in tables or graphs!

  9. New Sequences The previous sequences were the only ones taught in Algebra 2. But, it is possible for a sequence to be neither arithmetic nor geometric. Example: Find a formula for the general term of the sequence below n=1 n=2 n=3 n=4 n=5

  10. White Board Challenge Example: Find a formula for the general term of the sequence below n=1 n=2 n=3 n=4 n=5

  11. Monotonic Sequence A sequence is monotonic if it is either increasing (if for all ) or decreasing (if for all ). Example 1: Find the first 4 terms of to see how the sequence is monotonic.

  12. Example 2 Prove the sequence is decreasing. If the sequence is decreasing, for all . IF: THEN: Since the denominator is smaller: OR Therefore, is decreasing.

  13. Bounded Sequence A sequence is bounded above if there is a number such that for all A sequence is bounded below if there is a number such that for all If it is bounded above and below, then is a bounded sequence.

  14. Example Determine if the sequences below bounded below, bounded above, or bounded. Therefore, is bounded below. Since : 1 Since : The sequence is not bounded above. Since : >0 Therefore, is bounded. Since : <1.

  15. Limit of a Sequence A sequence has the limit and we write: or as if we can make the terms as close to as we like by taking sufficiently large. If exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).

  16. Reminder: Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

  17. Reminder: Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

  18. Example Determine if the sequences below converge or diverge. If the sequence converges, find its limit. Converges to 1 Diverges Converges to 0

  19. White Board Challenge Determine whether the sequence converges or diverges. If it converges, find its limit. Converges to

  20. Absolute Value Theorem It is not always possible to easily find the limit of a sequence. Consider: The Absolute Value Theorem states: If , then .

  21. Example Determine if the sequences below converge or diverge. Because of the Absolute Value Theorem, Converges to 0 Since the limit does not equal 0, we can not apply the Absolute Value Theorem. It does not mean it diverges. Another test is needed. The sequence diverges since it does not have a limit: -1,1,-1,1,-1,…

  22. Theorem: Bounded, Monotonic Sequences Every bounded, monotonic sequence is convergent. Example: Investigate the sequence below. Since the sequence appears to be monotonic and bounded, it appears to converge to 6. The sequence appears to be monotonic: It is increasing. The sequence appears to be bounded: The limit of the sequence appears to be 6.

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