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GeometricSequences and
Series
1, 4, 7,10,13
9,1, 7, 15
6.2, 6.6, 7, 7.4
, 3, 6
Arithmetic Sequences
ADDTo get next term
2, 4, 8,16, 32
9, 3,1, 1/ 3
1,1/ 4,1/16,1/ 64
, 2.5 , 6.25
Geometric Sequences
MULTIPLYTo get next term
Arithmetic Series
Sum of Terms
35
12
27.2
3 9
Geometric Series
Sum of Terms
62
20 / 3
85 / 64
9.75
2. Geometric Sequences and Series
a1 a2 a3 a4 a5 a6anan - 1
+ d + d + d + d + d
r
+ d
r r r r r
Geometric Sequences (Type 2)• In geometric sequences, you multiply by a common
ratio (r) each time.
•1, 2, 4, 8, 16, ... multiply by 2•27, 9, 3, 1, 1/3, ...Divide by 3 which means multiply by 1/3
ie ru
u
n
n 1
The nth term of an geometric sequence is denoted by the formula
1 nn aru
Where a is the 1st term and r is the common ratio
The sum of the first n terms of a geometric series is found by using:
)()(
r
raS
n
n
1
1
Note if r>1 then we can use the formula
Which is more convenient )1(
)1(
r
raS
n
n
Vocabulary of Sequences (Universal)
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
n 1n 1
n
n 1
nth term of geometric sequence
sum of the first n terms of geometric seq
a a r
1 rue ce S a
1n
r
1 n
n
or
sum of the first n terms of geometric sequenca ra
S1
er
Find the next three terms of 2, 3, 9/2, ___, ___, ___
3 – 2 vs. 9/2 – 3… not arithmetic3 9 / 2 3
1.5 geometric r2 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
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
2/3
n 1n 1a a r
9 1
9
1 2a
2 3
8
9 8
2a
2 3
7
8
2
3
128
6561
Example 2
Find u10 for the geometric sequence 144, 108, 81, 60¾, …
Answer
a = 144 and r = 4
3
12
9
144
108
u10 = arn-1
= 144 (¾)9 = 10.812…
Example 3
Find S19 for the geometric sequence 3-6+12-24+…
Answer
a = 3 and r = 23
6
5242893
5242893
3
52428813
21
213
1
1 19
)())((
)())((
)()(
r
raS
n
n
2 4 1
2Find a a if a 3 and r
3
-3, ____, ____, ____
2Since r ...
3
4 83, 2, ,
3 9
2 4
8 10a a 2
9 9
9Find a of 2, 2, 2 2,...
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
x
9
NA
2
2 2 2r 2
22
n 1n 1a a r
9
9
1
a 2 2
8
2 2
16 2
5 2 4If a 32 2 and r 2, find a to a .
____, , ____,________ ,32 2
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
x
5
NA
32 2
2n 1
n 1a a r
5
1
1
32 2 a 2
4
132 2 a 2
132 2 4a
1a 8 2
2a 8 2( 2) 16
23a 8 2( 2) 16 2
34a 8 2( 2) 32
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
n
n 1
1 rS a
1 r
7
7
11
1s
2 1
212
71
1 22
1
12
7
11
2
63
64
1, 4, 7, 10, 13, …. Infinite Arithmetic No Sum
3, 7, 11, …, 51 Finite Arithmetic
n 1 n
1
nS a a
2n 1
n a d2
1, 2, 4, …, 64 Finite Geometric
n1 n
n 1
a ra1 rS a
1 r 1 r
1, 2, 4, 8, … Infinite Geometricr 1 or r -1
No Sum
1 1 13,1, , , ...
3 9 27Infinite Geometric
-1 < r < 11a
S1 r
13.5 Infinite Geometric Series
|r| 1
|r| < 1
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
Find the sum, if possible: 2 2 8 16 2 ...
8 16 2r 2 2
82 2 r 1 No
NO SUM
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
Find the sum, if possible: 2 4 8
...7 7 7
4 87 7r 22 47 7
r 1 No
NO SUM
Find the sum, if possible: 5
10 5 ...2
55 12r
10 5 2 1 r 1 Yes
1a 10S 20
11 r 12
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
32/5
40
32
32/5
40S 45
504
10
1554
0S 2 5 500 50
504
15
450
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
An old grandfather clock is broken. When the pendulum is swung it follows a swing pattern of 25 cm, 20 cm, 16 cm, and so on until it comes to rest. What is the total distance the pendulum swings before coming to rest?
25
20
16
25
20
16
S
254
15
2 250