Geometric Sequences

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