1. Geometric Sequences 03-31-16

2. Types of sequences When you are repeatedly adding or subtracting the same value to/from the previous number to get the next number it is called an arithmetic sequence. For example: 3, 11, 19, 27, 35,… is an arithmetic sequence. When you are repeatedly multiplying or dividing by the same value to get the next number, it is called an geometric sequence. For example: 5, 25, 125, 625, … is a geometric sequence.

3. Find the next three terms… 1, 4, 16, 64,… 5, –10, 20,–40,… 512, 384, 288,…

4. The variable a is often used to represent terms in a sequence. The variable a 4 (read “a sub 4”)is the fourth term in a sequence. Writing Math

5. If the first term of a geometric sequence is a 1, the nth term is a n, and the common ratio is r, then an = a1rn–1an = a1rn–1 nth term1 st termCommon ratio Geometric Sequence Explicit Formula

6. The first term of a geometric sequence is 500, and the common ratio is 0.2. What is the 7th term of the sequence? a n = a 1 r n–1 Write the formula. a 7 = 500(0.2) 7–1 Substitute 500 for a 1,7 for n, and 0.2 for r. = 500(0.2) 6 Simplify the exponent. = 0.032 Use a calculator. The 7th term of the sequence is 0.032. Finding the nth Term of a Geometric Sequence

7. Examples: For a geometric sequence, a 1 = 5, and r = 2. Find the 6th term of the sequence. What is the 9th term of the geometric sequence 2, –6, 18, –54, …? a n = a 1 r n–1

8. The table shows a car’s value for 3 years after it is purchased. The values form a geometric sequence. How much will the car be worth in the 10th year? YearValue ($) 110,000 28,000 36,400 10,000 8,000 6,400 The value of r is 0.8. a n = a 1 r n–1 Example:

9. a n = a 1 r n–1 Write the formula. a 6 = 10,000(0.8) 10–1 Substitute 10,000 for a 1, 10 for n, and 0.8 for r. = 10,000(0.8) 9 Simplify the exponent. = 1,342.18 Use a calculator. In the 10th year, the car will be worth $1342.18. Example

10. Find the next three terms in each geometric sequence. 1. 3, 15, 75, 375,… 2. 3. The first term of a geometric sequence is 300 and the common ratio is 0.6. What is the 7th term of the sequence? 4. What is the 15th term of the sequence 4, –8, 16, –32, 64…? Try these… a n = a 1 r n–1

11. 5. The table shows a car’s value for three years after it is purchased. The values form a geometric sequence. How much will the car be worth after 8 years? YearValue ($) 118,000 215,300 313,005 Try these (cont)…

12. P. 249 #9-12, 17-20, 22, 24, & 26

13. Geometric Sequences Recursive Form 04-01-16

14. If the first term of a geometric sequence is a 1, the nth term is a n, and the common ratio is r, then a n = a(n-1)r ; for n≥2 nth term1 st termCommon ratio Geometric Sequence Recursive Form a 1 = a Initial condition or starting value

15. Find the recursive form for the sequence 7, 21, 63, 18, …