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Today in Precalculus

Learn about sequences in precalculus, including arithmetic and geometric types, explicit and recursive definitions, and general formulas for finding terms. Practice problems included. Homework: Complete assigned problems.

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Today in Precalculus

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  1. Today in Precalculus • Notes: Sequences • Homework • Go over quiz

  2. Vocabulary and notation • Sequences: an ordered progression of numbers. • Term: each number in a sequence is a term First term is a1 Second term is a2 nth term is an The subscripts denote only the position of the term in the sequence.

  3. Types • Arithmetic Sequence: a sequence in which there is a common difference, d, between every pair of successive terms. Example: 5,8,11,14 • Geometric: a sequence in which there is a common ratio, r, between every pair of successive terms. Example:

  4. Types • Infinite: there is an infinite number of terms in the sequence Example: • Finite: a finite number of terms in the sequence. Example: 5,8,11,14 • Sequences are infinite unless otherwise specified.

  5. Explicitly Defined Sequence • A formula is given for any term in the sequence Example: ak = 2k - 5 Find the first 5 terms and the 20th term for the sequence a1 = 2(1) – 5 = – 3 a2 = 2(2) – 5 = – 1 a3 = 2(3) – 5 = 1 a4 = 2(4) – 5 = 3 a5 = 2(5) – 5 = 5 a20 = 2(20) – 5 = 35

  6. Recursively Defined Sequence • The first term is given and along with a rule to obtain each succeeding term from the one preceding it. Example: b1 = 8 and bn = bn-1 – 2 for all n>1 Find the next 4 terms for the sequence b2 = b1 – 2 = 8 – 2 = 6 b3 = 6 – 2 = 4 b4 = 4 – 2 = 2 b5 = 2 – 2 = 0

  7. General formulas for finding terms in a sequence • Arithmetic: an = a1 + (n – 1)d • Geometric: an = a1r(n–1) • To use these: 1) Determine if the sequence is arithmetic or geometric 2) Find the common difference or ratio

  8. Example 1 • Find the 20th term of the sequence 55,49,43, … and write a recursive and explicit rule. • Arithmetic sequence with d= -6 • a20 = 55 + (20 – 1)(-6) a20 = –59 • Recursive rule: ak = ak-1 – 6 • Explicit rule: an = 55 + (n – 1)(-6) an = 55 – 6n + 6 an = 61 – 6n

  9. Example 2 • Find the 8th term of the sequence and write a recursive and explicit rule. • Geometric sequence with r=4 • Recursive rule: ak = 4ak-1 • Explicit rule:

  10. Practice Given the sequence 4, -8, 16, -32,…. Determine the type of sequence, state the recursive and explicit rules and state the 13th term.

  11. Practice Given the sequence 123, 96, 69,…. Determine the type of sequence, state the recursive and explicit rules and state the 13th term.

  12. Homework • Pg 739: 1-9odd, 21-31odd

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