1

Find the 9th term of the geometric sequence

7, 21, 63, . . .a1 = 7

The 9th term is 45,927.

21 37

r

an = a1rn – 1 = 7(3)n – 1

a9 = 7(3)9 – 1 = 7(3)8

= 7(6561) = 45,927

M3U1D4 Warm Up

NEW SEATS!!!!

Homework Check:

2831150

Homework Check:

Homework Check:Document Camera

Collect Parent Letters

M3U1D4 Geometric Series

Objective:To write arithmetic and geometric

sequences both recursively and with an explicit formula, use them to model situations, and translate between the

two forms. F-BF.2

First Let’s Look at a Geometric Finite Series

Vocabulary of Geometric Sequences and Series (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

n 1

n 1

n1

n

nth term of geometric sequence

sum of n terms of geometric sequ

a a r

a r 1S

r 1ence

r

raor

n

1

)1(1WRITE THESE FORMULAS DOWN ON YOUR HINTS

CARD!!!

FINITE!!!

1 9

1 2If a , r , find a .

2 3

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

1/2

x

9

NA, yet…

2/3

n 1n 1a a r

9 11 2

x2 3

8

8

2x

2 3

7

8

2

3 128

6561

And S9

Sn = )

32

1

)32

(1)(2/1(

1

)1(9

1

r

ra n

46098.1

Example 1: Let’s Review Geometric Sequences and add!!!

7

1 1 1Find S of ...

2 4 8

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

1/2

7

x

NA

11184r

1 1 22 4

n1

n

a r 1S

r 1

71 12 2

x12

1

1

71 12 2

12

1

127/128

Example 2:

Now Let’s Look at a Geometric Infinite Series

1, 4, 7, 10, 13, …. Infinite Arithmetic No Sum

3, 7, 11, …, 51 Finite Arithmetic n 1 n

nS a a

2

1, 2, 4, …, 64 Finite Geometric n

1

n

a r 1S

r 1

1, 2, 4, 8, … Infinite Geometricr > 1r < -1

No Sum

1 1 13,1, , , ...

3 9 27Infinite Geometric

-1 < r < 11a

S1 r

Let’s summarize what we know and add to it!

Find the sum, if possible: 1 1 1

1 ...2 4 8

1 112 4r

11 22

1 r 1 Yes

1a 1S 2

11 r 12

Example 3:

Find the sum, if possible: 2 2 8 16 2 ...

8 16 2r 2 2

82 2 1 r 1 No

NO SUM

Example 4:

Find the sum, if possible: 2 1 1 1...

3 3 6 12

1 113 6r

2 1 23 3

1 r 1 Yes

1

2a 43S

11 r 312

Example 5:

Find the sum, if possible: 2 4 8...

7 7 7

4 87 7r 22 47 7

1 r 1 No

NO SUM

YOU TRY!

Find the sum, if possible: 510 5 ...

2

55 12r

10 5 2 1 r 1 Yes

1a 10S 20

11 r 12

YOU TRY!

Find n for the series in which 1 na 5, d 3, S 440

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

5

x

y

440

3

n 1a a n 1 d

n 1 n

nS a a

2

y 5 31x

x40 y4

25

12

x440 5 5 x 3

x 7 x440

2

3

880 x 7 3x 20 3x 7x 880

X = 16

Graph on positive window

Example 6:

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?

1 20 1 19d c

1 201 20 19 1 39na a n d a

20

2020 39 10 59 590

2S

Example 7: A Real World Example!

(n-1)

The Bouncing Ball Problem – Version A

A ball is dropped from a height of 50 feet. It rebounds 4/5 of

it’s height, and continues this pattern until it stops. How far

does the ball travel?50

40

32

25.6

40

32

25.6

40S 45

504

10

1554

Example 8:

Notice…Drawing a Picture made visualizing and solving EASIER!!!

The Bouncing Ball Problem – Version B

A ball is thrown 100 feet into the air. It rebounds 3/4 of

it’s height, and continues this pattern until it stops. How far

does the ball travel?

100

75

225/4

100

75

225/4

10S 80

100

4 43

1

0

10

3

YOU TRY!

REMEMBER…Drawing a Picture made visualizing & solving easier!!!

Finally, how can we evaluate these in the calculator???

Graphing Utility: Find the first 5 terms of the geometric sequence an = 2(1.3)n.

List Menu:

variable

Graphing Utility: Find the finite sum 10

1

22 .3

n

n

List Menu:

beginning value

end value

variable

upper limit

lower limit

ProjectUPDATE THE PERCENTAGES AS FOLLOWS:

Don’t forget….

DUE Sept. 15th !!!

See me personally if you’ve already started calculations

Classwork:U1D4 Back Side WS odds

#1 reads:

Homework:Finish CW