1. Geometric Sequences and Series Unit 10.3

2. Practical Application “The company has been growing geometrically”

3. Purpose 1.Calculate for a geometric sequence 2.Calculate for the nth term 3.Compute for geometric means 4.Calculate for sums of geometric series 5.Calculate for sum in sigma notation

4. Calculate for geometric sequence 1.4, 11, 30.25 What are the next 4 in the series? Find common Ratio - Second term divided by first term 11 /4 = 2.75: common ratio is 2.75 30.25 * 2.75 = 83.19 83.19 * 2.75 = 228.77 228.77 * 2.75 = 629.11

5. Page 615 Problems 1 - 6 Find the common ratio and the next 3 terms 1.Common ratio = -2 2, -4, 8 2.Common ratio = -.75 - 27/128 81/512 -243/2,048 3.Common ratio = 1.5 1.69, 2.53, 3.8 4.Common ratio = 2.5 125, 312.5, 781.25 5.Common ratio = 5 250x, 1250x, 6250x 6.Common ratio = ¼ x, 1/4x, 1/16x

6. Formula a n = a 1 r n – 1 a n = final term in sequence a 1 = first term in sequence r = ratio n = term in sequence

7. Example 1.Find the 9 th term of the geometric sequence 4, 14, 49 1.Find the ratio 14/4 = 7/2 or 3.5 2.a n = 4(3.5) 8 3.a n = 4(22518.75) 4.a n = 90,075.8

8. Problems Page 615 Problems 20 - 24

9. Geometric Means Write a sequence that has two geometric means between 480 and -7.5 Step 1. 480, a, b, -7.5 (4 terms) Step 2. a 4 = a 1 r n-1 -7.5 = 480r 4-1 -7.5 = 480r 3 -7.5/480 = r 3 -1/64 = r 3 -1/4 = r a 2 =-120 =(480*-.25) a 3 =30 = (-120*-.25)

10. Problems Unit 10.3 Page 615 Problems 32 - 36

11. Geometric Series Geometric Sequence 2, 4, 8, 16…. Geometric Series is the sum of the terms of a geometric sequence 2 + 4 + 8 + 16… Formulas S n = a 1 (1 – r n )/(1 – r) S n = a 1 – a n r (1 – r)

12. Explanation of Formulas a) S n = a 1 (1 – r n )/(1 – r) b) S n = a 1 – a n r nth partial sum (1 – r)

13. Examples Page 612 Guided practice 6a Find the first 11 terms 7 + (-24.5) + 85.75 a 1 = 7 r = -24.5/7 = -3.5 s 11 = 7(1 – (3.5) 11 /(1 – (-3.5)) s 11 = 1,501,877.34

14. Examples Page 612 Guided practice 6b Find the sum of the first n terms a 1 = -8 a n = 131,072 r = -4 S n = a 1 – a n r 1 – r = -8 - (131,072)(-4)= 104,856 1 – (-4)

15. Geometric Sum in Sigma Notation 7 ∑ 3(5) n – 1 n = 2 See board

16. Sum of an infinite Geometric Series S = a 1 1 – r Problem: Find the sum of 10, -5, 2.5 r = -.5, a 1 = 10 S = 10/(1 – (-.5) = 6.67

17. Problems Page 605-6 Problems 40 – 46, 48 – 52, 56,57