1. 11.3 – Geometric Sequences

2. What is a Geometric Sequence?  In a geometric sequence, the ratio between consecutive terms is constant. This ratio is called the common ratio.  Unlike in an arithmetic sequence, the difference between consecutive terms varies.  We look for multiplication to identify geometric sequences.

3. Ex: Determine if the sequence is geometric. If so, identify the common ratio  1, -6, 36, -216 yes. Common ratio=-6  2, 4, 6, 8 no. No common ratio no. No common ratio

4. Important Formulas for Geometric Sequence:  Recursive Formula  Explicit Formula a n = (a n – 1 ) r a n = a 1 * r n-1 Where: a n is the nth term in the sequence a 1 is the first term n is the number of the term r is the common ratio  Geometric Mean Find the product of the two values and then take the square root of the answer.

5. Let’s start with the geometric mean  Find the geometric mean between 3 and 48 Let ’ s try one: Find the geometric mean between 28 and 5103

6. Ex: Write the explicit formula for each sequence First term: a 1 = 7 Common ratio = 1/3 Explicit: a n = a 1 * r n-1 Now find the first five terms: a 1 = 7(1/3) (1-1) = 7 a 2 = 7(1/3) (2-1) = 7/3 a 3 = 7(1/3) (3-1) = 7/9 a 4 = 7(1/3) (4-1) = 7/27 a 5 = 7(1/3) (5-1) = 7/81

7. Explicit Arithmetic Sequence Problem Find the 19 th term in the sequence of 11,33,99,297... a 19 = 11(3) 18 =4,261,626,379 Common ratio = 3 a 19 = 11 (3) (19-1) Start with the explicit sequence formula Find the common ratio between the values. Plug in known values Simplify a n = a 1 * r n-1

8. Student Check Find the 10 th term in the sequence of 1, -6, 36, -216... a 10 = 1(-6) 9 = -10,077,696 Common ratio = -6 a 10 = 1 (-6) (10-1) Start with the explicit sequence formula Find the common ratio between the values. Plug in known values Simplify a n = a 1 * r n-1

9. Word Problems:  If the number of bacteria in a colony doubles every five days and on Jan. 16 th there were 8,000,000 bacteria, how many bacteria were there on Jan 6 th ?

10. Word Problem  A women made $35,000 on the first year of her new job. Each year she receives a 10% raise. How much did she earn on the tenth year of her job?