2-3. arithmetic and geometric sequences

8/18/2019 2-3. Arithmetic and Geometric Sequences

1/46

9.2 – Arithmetic Sequences and Series


2/46

An introduction…………

1, 4, 7,10,13

9,1, 7, 15

6.2, 6.6, 7, 7.4

, 3, 6

− −

π π + π +

Arithmetic Sequences

ADD

To get next term

2, 4, 8,16, 32

9, 3,1, 1/ 3

1,1/ 4,1/16,1/ 64

, 2.5 , 6.25

− −

π π π

Geometric Sequences

MULTIPLY

To get next term

Arithmetic Series

Sum of Terms

35

12

27.2

3 9

−

π +

eometric Series

Sum of Terms

62

20/3

85/64

9.75π


3/46

!ind the ne"t four terms of #9, $2, 5, …

Arithmetic Se%uence

2 9 5 2 7− − − = − − =7 is referred to &s the common difference 'd(

)ommon *ifference 'd( # +h&t +e A** to et ne"t term

-e"t four terms……12, 19, 26, 33


4/46


5/46

oc&u&r of Se%uences 'niers&(

1& !irst term→

n& nth term→

nS sum of n terms→

n numer of terms→

d common difference

→( )

( )

n 1

n 1 n

nth term of &rithmetic se%uence

sum of n terms of &rithmetic se%uen

& & n 1 d

nS & &

2

ce

= + −

= +

→

→


6/46

ien &n &rithmetic se%uence +ith 15 1& 38 &nd d 3, find & .= = −

1& !irst term→

n& nth term→


n numer of terms→

d common difference→

"

15

38

-A

$3

( )n 1& & n 1 d= + −

( ) ( )38 " 1 15 3= + − −

80


7/46

63!ind S of 19, 13, 7,...− − −

1& !irst term→

n& nth term→


n numer of terms→


$19

63

"

6

( )n 1& & n 1 d= + −

( ) ( ) 19 6 1 353

3 6= + −=

−

353

( )n 1 nn

S & &2

= +

( )63 63 3 3S2

19 5−= +

63 1 1S 052=


8/46


9/46

n 1!ind n if & 633, & 9, &nd d 24= = =

1& !irst term→

n& nth term→


n numer of terms→


9

"

633

-A

24( )n 1& & n 1 d= + −

( )633 9 21" 4= + −

633 9 2 244"= + − 27


10/46

1 29!ind d if & 6 &nd & 20= − =

1& !irst term→

n& nth term→


n numer of terms→


$6

29

20

-A

"( )n 1& & n 1 d= + −

( )120 6 29 "= + −−

26 28"=13

"14

=


11/46

!ind t+o &rithmetic me&ns et+een #4 &nd 5

$4, , , 5

1& !irst term→n& nth term→


n numer of terms→


$4

4

5

-A"

( )n 1& & n 1 d= + −

( ) ( )15 4 4 "= + −−" 3=

The t+o &rithmetic me&ns &re #1 &nd 2, since #4, $1, 2, 5

forms &n &rithmetic se%uence


12/46

!ind three &rithmetic me&ns et+een 1 &nd 4

1, , , , 4



n numer of terms→


1

5

4

-A"

( )n 1& & n 1 d= + −

( ) ( )4 1 "15= + −3

"4

=

The three &rithmetic me&ns &re 7/4, 10/4, &nd 13/4

since 1, 7/4, 10/4, 13/4, 4 forms &n &rithmetic se%uence


13/46

!ind n for the series in +hich 1 n& 5, d 3, S 440= = =

1& !irst term→

n& nth term→


n numer of terms→


5

"

440

3

( )n 1& & n 1 d= + −

( )n 1 nnS & &2

= +

( ) 5 31"= + −

( )"

40 42

5= +

( )( )12

"440 5 5 " 3= + + −

( )" 7 "440

2

3=

+

( )880 " 7 3"= +20 3" 7" 880= + −

16

r&:h on :ositie +indo+


14/46

The sum of the first n terms of an infinite sequence

is called the nth partial sum.

1(

2n n

nS a a= +


15/46

Example !. "ind the 1#$th partial sum of the arithmetic sequence% #%

1!% 2&% '(% )9% *

1 # 11 # 11 !a d c= = → = − = −

11 !n

a n= − ( )1#$ 11 1#$ ! 1!))a→ = − =

( ) ( )1#$1#$

# 1!)) 1!)9 12'%!2

S = + = =


16/46

Example &. An auditorium has 2$ ro+s of seats. There are 2$ seats in

the first ro+% 21 seats in the second ro+% 22 seats in the third ro+% and

so on. ,o+ man- seats are there in all 2$ ro+s

1 2$ 1 19d c= = − =

( ) ( )1 2$1 2$ 19 1 '9na a n d a= + − → = + =

( ) ( )2$2$

2$ '9 1$ #9 #9$2

S = + = =


17/46


18/46

9.' – eometric Sequences and Series


19/46

1, 4, 7,10,13

9,1, 7, 15

6.2, 6.6, 7, 7.4

, 3, 6

− −

π π + π +

Arithmetic Sequences

ADD

To get next term

2, 4, 8,16, 32

9, 3,1, 1/ 3

1,1/ 4,1/16,1/ 64

, 2.5 , 6.25

− −

π π π

Geometric Sequences

MULTIPLY

To get next term

Arithmetic Series

Sum of Terms

35

12

27.2

3 9

−

π +

eometric Series

Sum of Terms

62

20/3

85/64

9.75π


20/46

oc&u&r of Se%uences 'niers&(

1& !irst term→

n& nth term→


n numer of terms→

r common r&tio→

( )

n 1

n 1

n

1

n

nth term of eometric se%uence

sum of n terms of eometric se%u

& & r

& r 1S

r 1

ence

−→ =

− =

−

→


21/46

!ind the ne"t three terms of 2, 3, 9/2, , ,

3 # 2 s. 9/2 # 3… not &rithmetic

3 9 / 2 31.5 eometric r 2 3 2= = → → =

3 3 3 3 3 3

2 2 2

92, 3, , , ,

2

9 9 9

2 2 2 2 2 2

× × × × × ×

92, 3, , ,

27 81 243

4 8,

2 16


22/46

1 9

1 2;f & , r , find & .

2 3= =

1& !irst term→

n& nth term→


n numer of terms→

r common r&tio→

1/2

"

9

-A

2/3

n 1

n 1& & r −=

9 11 2

"2 3

− =

8

8

2"

2 3=

×

7

8

2

3= 128

6561=


23/46

!ind t+o eometric me&ns et+een #2 &nd 54

$2, , , 54

1& !irst term→

n& nth term→


n numer of terms→

r common r&tio→

$2

54

4

-A"

n 1

n 1& & r −=

( ) ( )14

54 2 " −

−=3

27 "− =3 "− =

The t+o eometric me&ns &re 6 &nd $18, since #2, 6, $18, 54

forms &n eometric se%uence


24/46

2 4 1

2!ind & & if & 3 &nd r

3− = − =

$3, , ,

2Since r ...

3=

4 83, 2, ,

3 9

− −− −

2 4

8 10

& & 2 9 9

− −

− = − − =


25/46

9!ind & of 2, 2, 2 2,...

1& !irst term→

n& nth term→


n numer of terms→

r common r&tio→

"

9

-A

2

2 2 2r 2

22= = =

n 1

n 1& & r −=

( ) 9 1

" 2 2−

=

( )8

" 2 2=

" 16 2=


26/46

5 2;f & 32 2 &nd r 2, find &= = −

, , , ,32 2



n numer of terms→

r common r&tio→

"

5

-A

32 2

2−n 1

n 1& & r −=

( )

5 1

32 2 " 2−

−=

( )4

32 2 " 2= −

32 2 "4=

8 2 "=


27/46


28/46

7

1 1 1!ind S of ...

2 4 8+ + +

1& !irst term→

n& nth term→


n numer of terms→

r common r&tio→

1/2

7

"

-A

11

184r 1 1 2

2 4

= = =

( )n1n

& r 1S

r 1

− =−

71 1

2 2"

1

2

1

1

− =−

71 1

2 2

1

2

1 − =

−

63

64=


29/46

Section 12.3 # ;nfinite Series


30/46

1, 4, 7, 10, 13, …. ;nfinite Arithmetic -o Sum

3, 7, 11, …, 51 !inite Arithmetic ( )n 1 nn

S & &

2

= +

1, 2, 4, …, 64 !inite eometric ( )n1

n

& r 1S

r 1

−=

−

1, 2, 4, 8, … ;nfinite eometric

r > 1

r ? $1

-o Sum

1 1 13,1, , , ...3 9 27

;nfinite eometric$1 ? r ? 1

1&S1 r

= −


31/46

!ind the sum, if :ossie1 1 1

1 ...2 4 8

+ + + +

1 1

12 4r 11 2

2

= = = 1 r 1 @es→ − ≤ ≤ →

1& 1S 211 r

12

= = =− −


32/46

!ind the sum, if :ossie 2 2 8 16 2 ...+ + +

8 16 2r 2 2

82 2= = = 1 r 1 -o→ − ≤ ≤ →

- SB


33/46

!ind the sum, if :ossie2 1 1 1

...3 3 6 12

+ + + +

1 1

13 6r 2 1 2

3 3

= = = 1 r 1 @es→ − ≤ ≤ →

1

2& 43S

11 r 31

2

= = =− −

2 4 8


34/46

!ind the sum, if :ossie2 4 8

...7 7 7

+ + +

4 8

7 7r 22 4

7 7

= = = 1 r 1 -o→ − ≤ ≤ →

- SB


35/46

!ind the sum, if :ossie5

10 5 ...2

+ + +

5

5 12r 10 5 2

= = = 1 r 1 @es→ − ≤ ≤ →

1& 10S 2011 r

12

= = =− −


36/46

The Councin C& Droem # ersion A

A & is dro::ed from & heiht of 50 feet. ;t reounds 4/5 of

itEs heiht, &nd continues this :&ttern unti it sto:s. Fo+ f&r

does the & tr&e50

40

32

32/5

40

32

32/5

40S 45

50

41

0

155

4= =

−+

−


37/46

The Councin C& Droem # ersion C

A & is thro+n 100 feet into the &ir. ;t reounds 3/4 of

itEs heiht, &nd continues this :&ttern unti it sto:s. Fo+ f&r

does the & tr&e

100

75

225/4

100

75

225/4

10S 80

100

4 4

31

0

1

03

= =−

+−


38/46

Sim& -ot&tion


39/46

C

n

n A&

=∑

DDGH C-*

'-BCGH(

IJGH C-*

'-BCGH(

S;BA'SB ! TGHBS( -TF TGHB

'SGKG-)G(

4


40/46

( ) L

4

1

L 2=

+∑ ( )21= + ( )2 2+ + ( )3 2+ + ( )24+ + 18=

( )7

4&

2&=∑ ( )( )42= ( )( )2 5+ ( )( )2 6+ ( )( )72+ 44=

( )nn 0

4

0.5 2=

+

∑ ( )00.5 2= + ( )10.5 2+ + ( )20.5 2+ + ( )30.5 2+ + ( )

40.5 2+ +

33.5=

n 0 1 2


41/46

0

n

36

5=

∞ =

∑0

36

5

13

65

+

23

65

+ ...+

1&S1 r = − 6 153

15

= =−

( )2

"

3

7

2" 1=

+∑ ( )( ) ( )( ) ( )( ) ( )( )2 1 2 8 1 2 9 1 ...7 2 123= + + + + + + + +

( ) ( )n 1 n2n 1

S & & 152

3

2

747

− += + = + 527=

19


42/46

( )1

9

4

4 3=

+∑ ( )( ) ( )( ) ( )( ) ( )( )4 3 4 5 3 4 6 3 ...4 4 319= + + + + + + + +

( ) ( )n 1 n1n 1

S & & 192

9

2

479

− +

= + = + 784=

H it i i t ti 3 6 9 12


43/46

He+rite usin sim& not&tion 3 M 6 M 9 M 12

Arithmetic, d 3

( )n 1& & n 1 d= + −( )n& 3 n 1 3= + −

n& 3n=

4

1n

3n=

∑

H i i i i 16 8 4 2 1


44/46

He+rite usin sim& not&tion 16 M 8 M 4 M 2 M 1

eometric, r N

n 1

n 1& & r

−

= n 1n

1& 16

2

− =

n 1

n

5

1

116 2

−

= ∑

H it i i t ti 19 18 16 12 4


45/46

He+rite usin sim& not&tion 19 M 18 M 16 M 12 M 4

-ot Arithmetic, -ot eometric

n 1

n& 20 2 −= −

n 1

n

5

1

20 2 −

=−∑

19 M 18 M 16 M 12 M 4 $1 $2 $4 $8

3 9 27


46/46

He+rite the foo+in usin sim& not&tion3 9 27

...5 10 15

+ + +

-umer&tor is eometric, r 3

*enomin&tor is &rithmetic d 5

-BGHATH ( )n 1

n3 9 27 ... & 3 3

−

+ + + → =

*G-B;-ATH ( )n n5 10 15 ... & 5 n 1 5 & 5n+ + + → = + − → =

S;BA -TAT;- ( )

1

1

n

n 5n

3 3 −∞

=∑

2-3. arithmetic and geometric sequences

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