Unit 4 Part B GEOMETRIC SEQUENCES These are sequences where the ratio of successive terms of a sequence is always the same number. This number is called the common ratio.

An introduction………… Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term

Ex: Determine if the sequence is geometric 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 This is an Arithmetic Sequence with “common difference” of 2

Important Formula for Geometric Sequence: an = a1  r n-1 Where: an is the nth term in the sequence a1 is the first term n is the number of the term r is the common ratio

Ex: Write the first 5 terms of this sequence with: First term: a1 = 7 Common ratio = 1/3 an = a1 * r n-1 a1 = 7(1/3) (1-1) = 7 a2 = 7(1/3) (2-1) = 7/3 a3 = 7(1/3) (3-1) = 7/9 a4 = 7(1/3) (4-1) = 7/27 a5 = 7(1/3) (5-1) = 7/81 Now find the first five terms:

Geometric Sequence Problem Find the 19th term in the sequence of 11,33,99,297 . . . an = a1 * r n-1 Start with the sequence formula Find the common ratio between the values. Common ratio = 3 a19 = 11 (3) (19-1) Plug in known values a19 = 11(3)18 =4,261,626,379 Simplify

Find the 10th term in the sequence of 1, -6, 36, -216 . . . Let’s try one Find the 10th term in the sequence of 1, -6, 36, -216 . . . an = a1 * r n-1 Start with the sequence formula Find the common ratio between the values. Common ratio = -6 a10 = 1 (-6) (10-1) Plug in known values a10 = 1(-6)9 = -10,077,696 Simplify

 2 r = 2 a = 1 1, 2, 4, 8, 16 . . . Try this to get the 5th term.

Find the 8th term of 0.4, 0.04. 0.004, . . . To find the common ratio, take any term and divide it by the term in front

Find the next four terms of –9, -2, 5, … Arithmetic Sequence 7 is referred to as the common difference (d) Common Difference (d) – what we ADD to get next term Next four terms……12, 19, 26, 33

Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of x, 2x, 3x, … Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -32k

Given an arithmetic sequence with x 38 15 NA -3 X = 80

Try this one: 1.5 16 x NA 0.5

9 x 633 NA 24 X = 27

-6 29 20 NA x

Example 7. An auditorium has 20 rows of seats Example 7. An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows?

Example 8. A small business sells $10,000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation. So the total sales for the first 2o years is $1,625,000

Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 – 2 vs. 9/2 – 3… not arithmetic

1/2 x 9 NA 2/3

x 9 NA