12.3 – Analyze Geometric Sequences and Series

Geometric Sequence: Ratio of any term to the previous term is constant Common Ratio: Ratio each term is increasing by

1. Tell whether the sequence is geometric. 4, 10, 18, 28, 40, … No

2. Tell whether the sequence is geometric. 625, 125, 25, 5, 1, … Yes

3. Tell whether the sequence is geometric. –4, 8, –16, 32, –64, … –2 Yes –2

Rule for a Geometric Sequence: The nth term of a geometric sequence with first term a 1 and common ratio r is given by:

4.Write a rule for the nth term of the sequence. Then find a 7. 4, 20, 100, 500, …. a 1 = 4 r = 5 a 7 =4(5) 7–1 a 7 = 4(5) 6 a 7 = 4(15625) a 7 = 62500

5. Write a rule for the nth term of the sequence. Then find a , –76, 38, –19, … a 1 = 152 r = 2 a 7 =152(-1/2) 7–1 a 7 = 152(-1/2) 6 a 7 = 152(1/64) a 7 = 19/8

6. Write a rule for the nth term of the geometric sequence, given: a 4 = 12, r = 2 12 = a 1 (2) 4 – 1 12 = a 1 (2) 3 12 = 8a 1 3/2 = a 1

7. Write a rule for the nth term of the geometric sequence, given: a 6 = –96, r = 2/3

8. Write a rule for the nth term of the geometric sequence, given: a 3 = –48, a 6 = 3072 r3r3

9. Write a rule for the nth term of the geometric sequence, given: a 2 = –12, a 4 = –3 r2r2

Sum of a Finite Geometric Series: The sum of the first n terms of a geometric series with common ratio r  1 is:

10. Find the sum of the geometric series. a 1 = 4(3) 1-1 = r = 3 4(1) =4 n = 16

11. Find the sum of the geometric series. a 1 = 12(-1/2) 0 = r = -1/2 12(1) =12 n = 8