Warm-Up

1.Grab a worksheet off the stool.

2.Complete all of the odds.

Homework Check

Geometric Sequencesand Series

What is a geometric sequence?

• In a geometric sequence, each term is found by multiplying the previous term by a constant.

What is a geometric sequence?

In general you write a Geometric Sequence like this:

{a, ar, ar2, ar3, ... }

Where a is the first term, and r is the factor between the terms called the common ratio.

Example: a = 1 r = 2

Term # How it is written with “r”

1st 1 1

2nd 2 1•2

3rd 4 1•22

4th 8 1•23

What do you notice? • Do you see a pattern between term

# and the exponent r is raised to?

• One less than the term number!

Term # How it is written with “r”

1st 2 2

2nd 4 2•21

3rd 8 2•22

4th 16 2•23

5th 32 2•24

6th 64 2•25

Formula

an = a1r(n-1)

Where an is the term we are looking for and a1 is the first term in the sequence…so if we want to find the third term (a3) we plug in 3 for n!

Look how we find each term…

Sequence

Term Formula a1r(n-1) Final Result

2 a1 a1r(1-1) = 2•2(1-1) 2•20 = 2•1 = 2

4 a2 a1r(2-1) = 2•2(2-1) 2•21 = 2•2 = 4

8 a3 a1r(3-1) = 2•2(3-1) 2•22 = 2•4 = 8

16 a4 a1r(4-1) = 2•2(4-1) 2•23 = 2•8 = 16

32 a5 a1r(5-1) = 2•2(5-1) 2•24 = 2•16 = 32

64 a6 a1r(6-1) = 2•2(6-1) 2•25 = 2•32 = 64

The graph….• We graph the order of the term for

“x” and the number in that spot for “y”

TermX

SequenceY

Point

1st 1 (1,1)

2nd 2 (2,2)

3rd 4 (3,4)

4th 8 (4,8)

5th 16 (5,16)

Examples

1) What is the first term, a1? What is the common ratio? Find the next 3 terms….

1, 3, 27, ___ , ___ , ___

2) What is the first term, a1? What is the common ratio? Find the next 3 terms….

70, 7, 0.7, ____ , ____ , ____

You Try

1) What is the first term, a1? What is the common ratio? Find the next 3 terms….

-3 , -6, -12, ___ , ___ , ___

2) What is the first term, a1? What is the common ratio? Find the next 2 terms….

8, -20, 50, -125, ____ , ____

Examples3) Set up a geometric

sequence with a1 = 5, and the common ratio of 2. Find the 4th term.

4) Set up a geometric sequence with a1 = 7 , and the common ratio of 1/5 . Find the 3rd term.

You Try3) Set up a geometric

sequence with a1 = 6, and the common ratio of 3. Find the 4th term using your formula.

4) Set up a geometric sequence with a1 = 2 , and the common ratio of ¼ . Find the 3rd term using your formula.

Examples5) Find the first, fourth, and

eighth term of each sequence.

an = 4•2(n-1)

6) Find the first, third, and sixth term of each sequence.

an = (0.5)•3(n-1)

You Try5) Find the first, fourth, and

eighth term of each sequence.

6) Find the first, fifth, and seventh term of each sequence.

an = -2 • 5(n-1) an = 0.25 • 3(n-1)

Decide if each sequence is geometric or arithmetic….

Sum of A Finite Geometric Series

• Finite- having limits, something that is measureable.

• Finite Series- the sum of the terms of a sequence.

• Geometric series-series in which the ratio of each two consecutive terms is a constant function of the sum.

Sum of A Finite Geometric Series

• Formula:

r

raS

n

n

1

)1(1

tio common ra

terms number oftermfirst

r

n a1

Example:

1. What is the sum of the first ten terms of the geometric series?

8 + 6 + 32 + 64 + 128 + ….

Example:

-10 – 5 – 2.5 – 1.25 - … ; n = 7

You Try:

3 + 12 + 48 + 192 + …; n = 6

Homework

• Workbook pg. 13 #1-9

Extra Credit: • Independent Practice Sheet:

Evens, on separate sheet of paper.

• Due Friday March 20.