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Geometric Sequences

Geometric sequences are a series of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This guide covers fundamental concepts of geometric sequences with examples illustrating how to find the nth term and calculate values within the sequence. It includes specific problem-solving strategies and homework instructions for practice. Key points include identifying terms, calculating common ratios, and applying geometric sequences in various contexts.

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Geometric Sequences

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  1. Geometric Sequences Choi 2012

  2. Geometric Sequence • A sequence like 3, 9, 27, 81,…, where the ratio between consecutive terms is a constant, is called a geometric sequence. In a geometric sequence, the first term t1, is denoted as a. Each term after the first is found by multiplying a constant, called the common ratio, r, to the preceding term. • The list then becomes . {a, ar, ar2, ar3,...}

  3. Geometric Sequences Formulas In general: {a, ar, ar2, ar3,...,arn-1 ,...}

  4. Example 1 – Finding Formula for the nth term In the geometric sequences: {5, 15, 45,...}, find a) b) c) c) b) a) 10 n 10 n n n 5 5

  5. Example 2 – Finding Formula for the nth term Given the geometric sequence: {3, 6, 12, 24, ...}. • Find the term • Which term is 384? We know the nth term is 384 !! b) a) n n 14 14 Drop the bases!!

  6. Example 3 – Find the terms in the sequence • In a geometric sequence, t3 = 20 and t6 = -540. Find the first 6 terms of the sequence. (2) (1) Substitute into (1) (1) (2) Therefore the first 6 terms of the sequences are:

  7. Example 4 – Find the terms in the sequence • In a geometric sequence, t3 = 20 and t6 = -540. Find the first 6 terms of the sequence. METHOD 2 To find a, we use the same thinking process!! t1 = 20r(1-3) tn=20r (n-3) Therefore the first 6 terms of the sequences are:

  8. Example 5 – Applications of Geometric sequence • Determine the value of x such that • Form a geometric sequence. Find the sequences and Therefore the sequences are: 5+4, 2(5)+5, 4(5)+5, ... 9, 15, 25,...

  9. Homework: • P. 453 #8, 11, 14 ,15,20 • P. 461 #1-12 • Course Pack: • Applications of Sequences

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