Geometric Sequences and Series

Unit 10.3

Practical Application

“The company has been growing geometrically”

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

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 term11 /4 = 2.75: common ratio is 2.7530.25 * 2.75 = 83.1983.19 * 2.75 = 228.77228.77 * 2.75 = 629.11

Page 615 Problems 1 - 6

Find the common ratio and the next 3 terms

1. Common ratio = -2 2, -4, 82. 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

Formula

an = a1rn – 1

an = final term in sequence

a1 = first term in sequence

r = ratio

n = term in sequence

Example

1. Find the 9th term of the geometric sequence 4, 14, 49

1. Find the ratio 14/4 = 7/2 or 3.5

2. an = 4(3.5)8

3. an = 4(22518.75)

4. an = 90,075.8

Problems

Page 615 Problems 20 - 24

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. a4 = a1rn-1

-7.5 = 480r4-1

-7.5 = 480r3 -7.5/480 = r3

-1/64 = r3

-1/4 = r a2 =-120 =(480*-.25) a3 =30 = (-120*-.25)

Problems

Unit 10.3 Page 615

Problems 32 - 36

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

Sn = a1(1 – rn)/(1 – r)

Sn = a1 – anr

(1 – r)

Explanation of Formulas

a) Sn = a1(1 – rn)/(1 – r)

b) Sn = a1 – anr nth partial sum

(1 – r)

Examples

Page 612 Guided practice 6a

Find the first 11 terms 7 + (-24.5) + 85.75

a1 = 7 r = -24.5/7 = -3.5

s11 = 7(1 – (3.5)11/(1 – (-3.5))

s11 = 1,501,877.34

Examples

Page 612 Guided practice 6b

Find the sum of the first n terms

a1 = -8 an = 131,072 r = -4

Sn = a1 – anr

1 – r

= -8 - (131,072)(-4) = 104,856

1 – (-4)

Geometric Sum in Sigma Notation

7

∑ 3(5)n – 1

n = 2

See board

Sum of an infinite Geometric Series

S = a1

1 – r

Problem: Find the sum of 10, -5, 2.5

r = -.5, a1 = 10

S = 10/(1 – (-.5) = 6.67

Problems

Page 605-6

Problems 40 – 46, 48 – 52, 56,57