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9.2 – Arithmetic Sequences and Series
An introduction…………
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
Find the next four terms of –9, -2, 5, …
Arithmetic Sequence
2 9 5 2 7
7 is referred to as the common difference (d)
Common Difference (d) – what we ADD to get next term
Next four terms……12, 19, 26, 33
Find the next four terms of 0, 7, 14, …
Arithmetic Sequence, d = 7
21, 28, 35, 42
Find the next four terms of x, 2x, 3x, …
Arithmetic Sequence, d = x
4x, 5x, 6x, 7x
Find the next four terms of 5k, -k, -7k, …
Arithmetic Sequence, d = -6k
-13k, -19k, -25k, -32k
Vocabulary of Sequences (Universal)
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
n 1
n 1 n
nth term of arithmetic sequence
sum of n terms of arithmetic sequen
a a n 1 d
nS a a
2ce
Given an arithmetic sequence with 15 1a 38 and d 3, find a .
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
x
15
38
NA
-3
n 1a a n 1 d
38 x 1 15 3
X = 80
63Find S of 19, 13, 7,...
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
-19
63
??
x
6
n 1a a n 1 d
?? 19 6 1
?? 353
3 6
353
n 1 n
nS a a
2
63
633 3S
219 5
63 1 1S 052
16 1Find a if a 1.5 and d 0.5 Try this one:
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
1.5
16
x
NA
0.5
n 1a a n 1 d
16 1.5 0.a 16 51
16a 9
n 1Find n if a 633, a 9, and d 24
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
9
x
633
NA
24
n 1a a n 1 d
633 9 21x 4
633 9 2 244x
X = 27
1 29Find d if a 6 and a 20
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
-6
29
20
NA
x
n 1a a n 1 d
120 6 29 x
26 28x
13x
14
Find two arithmetic means between –4 and 5
-4, ____, ____, 5
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
-4
4
5
NA
x
n 1a a n 1 d
15 4 4 x x 3
The two arithmetic means are –1 and 2, since –4, -1, 2, 5
forms an arithmetic sequence
Find three arithmetic means between 1 and 4
1, ____, ____, ____, 4
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
1
5
4
NA
x
n 1a a n 1 d
4 1 x15 3
x4
The three arithmetic means are 7/4, 10/4, and 13/4
since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence
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: The nth Partial Sum
The sum of the first n terms of an infinite sequence is called the nth partial sum.
1( )2n nnS a a
Example 6. Find the 150th partial sum of the arithmetic sequence, 5, 16, 27, 38, 49, …
1 5 11 5 11 6a d c
11 6na n 150 11 150 6 1644a
150
1505 1644 75 1649 123,675
2S
Example 7. 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 8. A small business sells $10,000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation.
1 10,000 7500 10,000 7500 2500a d c
1 201 10,000 19 7500 152,500na a n d a
20
2010,000 152,500 10 162,500 1,625,000
2S
So the total sales for the first 2o years is $1,625,000
9.3 – Geometric Sequences 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
Vocabulary of Sequences (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
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 11 2
x2 3
8
8
2x
2 3
7
8
2
3 128
6561
Find two geometric means between –2 and 54
-2, ____, ____, 54
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
-2
54
4
NA
x
n 1n 1a a r
1454 2 x
327 x 3 x
The two geometric means are 6 and -18, since –2, 6, -18, 54
forms an geometric sequence
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 1
x 2 2
8
x 2 2
x 16 2
5 2If a 32 2 and r 2, find 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
32 2 x 2
4
32 2 x 2
32 2 x4
8 2 x
*** Insert one geometric mean between ¼ and 4***
*** denotes trick question
1,____,4
4
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
1/4
3
NA
4
xn 1
n 1a a r
3 114
4r 2r
14
4 216 r 4 r
1,1, 4
4
1, 1, 4
4
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
63
64
Section 12.3 – 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
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 1 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
1 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
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
Sigma Notation
B
nn A
a
UPPER BOUND(NUMBER)
LOWER BOUND(NUMBER)
SIGMA(SUM OF TERMS) NTH TERM
(SEQUENCE)
j
4
1
j 2
21 2 2 3 2 24 18
7
4a
2a 42 2 5 2 6 72 44
n
n 0
4
0.5 2
00.5 2 10.5 2 20.5 2 30.5 2 40.5 2
33.5
0
n
b
36
5
0
36
5
13
65
23
65
...
1aS
1 r
6
153
15
2
x
3
7
2x 1
2 1 2 8 1 2 9 1 ...7 2 123
n 1 n
2n 1S a a 15
2
3
2
747
527
1
b
9
4
4b 3
4 3 4 5 3 4 6 3 ...4 4 319
n 1 n
1n 1S a a 19
2
9
2
479
784
Rewrite using sigma notation: 3 + 6 + 9 + 12
Arithmetic, d= 3
n 1a a n 1 d
na 3 n 1 3
na 3n4
1n
3n
Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1
Geometric, r = ½ n 1
n 1a a r n 1
n
1a 16
2
n 1
n
5
1
116
2
Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4
Not Arithmetic, Not Geometric
n 1na 20 2
n 1
n
5
1
20 2
19 + 18 + 16 + 12 + 4 -1 -2 -4 -8
Rewrite the following using sigma notation:3 9 27
...5 10 15
Numerator is geometric, r = 3Denominator is arithmetic d= 5
NUMERATOR: n 1
n3 9 27 ... a 3 3
DENOMINATOR: n n5 10 15 ... a 5 n 1 5 a 5n
SIGMA NOTATION: 1
1
n
n 5n
3 3