Algebra II
Unit 8
1
The table shows the cost of mailing letters first class in the U.S. in 1995.
Weight Not
Exceeding
(ounces)
Cost
1
$0.32
2
$0.55
3
$0.78
4
$1.01
5
$1.24
$0.32, $0.55, $0.78, $1.01, $1.24 This set of numbers is an example of a sequence.
Each number in the sequence is called a term.
The first term is symbolized by a1 , the second term by a 2 , and so on to a n , the nth term.
a1 = $0.32
a 2 = $0.55
a 3 = $0.78
a 4 = $1.01
a 5 = $1.24
You can find the next term in a sequence by looking for a pattern.
Example: Given the sequence: 3, 5, 7, 9, 11, ...
A. Find the next 3 terms of the sequence.
13, 15, 17
B. Find the rule for the nth term.
1 2 3 4 5 ••• n
3 5 7 9 11 • • • an
Notice each number in the sequence 3, 5, 7, 9, 11 is equal to one more than
twice the positive integer with which it is paired.
Rule: a n = 2n + 1
Algebra II
Unit 8
Use the rule to write the first three terms of each sequence:
1. a n = 5n
2. a n = n +
1
n
3. a n = n 2 + 1
4. an = −4n 2 − 2
€
Find the next 3 terms of the sequence.
Find the rule for the nth term.
5. 2, 4, 6, 8, ...
6. 4, 7, 10, 13, ...
7. 48, 45, 42, 39, ...
8. 2, 5, 8, 11, ...
9. -12, -7, -2 , 3, ...
10.
1 2 3 4
, , , ,...
2 3 4 5
2
Algebra II
Unit 8
Arithmetic Sequences
An arithmetic sequence is a sequence in which the difference between successive terms is a
constant.
The constant is called the common difference, d.
Formula:
An arithmetic sequence: -6, -2, 2, 6, 10, ... common difference is 4.
Example 1: The first term of an arithmetic sequence is 3 and the common difference is 2. Find
the twentieth term.
a n = a1 + (n − 1)d
a 2 0 = 3 + (20 − 1)2
a 2 0 = 3 + (19)2
a 2 0 = 3 + 38
a 2 0 = 41
1. Write the first five terms of each arithmetic sequence:
A. a1 =1, d=5
B. a1 =-5, d=6
C. a1 =-30, d=-2
3
−1
D. a1 = , d=
4
4
3
Algebra II
Unit 8
4
2. The first term of an arithmetic sequence is 16. The common difference is -2. Find a1 7 .
3. The first term of an arithmetic sequence is 2. The common difference is 5. Find a 6 .
4. In an arithmetic sequence a1 = 20 and d =
5
. Find a1 1.
2
5. In an arithmetic sequence a1 = −6 and d =
2
. Find a1 0 .
3
6. The first term of an arithmetic sequence is 2 and the common difference is 3. find the 30th
term.
Algebra II
Unit 8
7. The first term of an arithmetic sequence is 5 and the common difference is -4. find the
twenty-fifth term.
8. Find the 10th term of 4, 5, 6, ...
9. Find the 1000th term of 5, 7, 9, ...
10. Find the 200th term of 18, 14, 10, ...
11. The top row of a pile of logs contains 6 logs, the row below the top one contains 7 logs, the
third row from the top contains 8 logs, and so on. If there are 45 rows, how may logs are there
in the bottom row?
5
Algebra II
Unit 8
6
Example 2: The sixth term of an arithmetic sequence is -2 and the twelfth term is -14. find a1
and d.
___ ___ ___ ___ ___ - 2
1
-2
1
2
3
4
5
6
___ ___ ___ ___ ___ -14
7
8
9
10
11
12
___ ___ ___ ___ ___ -14
2
3
4
5
6
7
a n = a1 + (n − 1)d
-14 = -2+(7-1)d
-14 = -2+6d
-12 = 6d
d = -2
8
6
4
2
0
-2
-4
-6
-8
1
2
3
4
5
6
7
8
9
-10 -12 -14
10
11
12
Answer: d = -2 and a1 = 8
Example 3: The sixth term of an arithmetic sequence is -2 and the twelfth term is -14. find a1
and d. We are going to look at another way to do the same problem.
a 6 = a1 + (6 − 1)d
a1 2 = a 1 + (12 − 1)d
−2 = a1 + 5d
−14 = a1 + 11d
12 =
d = -2
-6d
Algebra II
Unit 8
7
8
6
4
2
0
-2
-4
-6
-8
1
2
3
4
5
6
7
8
9
-10 -12 -14
10
11
12
Answer: d = -2 and a1 = 8
Problems:
1. The fifth term of an arithmetic sequence is -8 and the twelfth term is -22. Find a1 and d.
2. The sixth term of an arithmetic sequence is 12 and the eleventh term is -3. Find a1 and d.
3. The fourth term of an arithmetic sequence is 13 and the sixth term is 7. Find the first term
and the common difference.
Algebra II
Unit 8
8
Arithmetic Means
The terms between any two given terms of an arithmetic sequence are called arithmetic means.
For example, in the sequence 2, 8, 14, 20, 26. ... the terms 8, 14, and 20 are the three arithmetic
means between 2 and 26.
Example 1: Find three arithmetic means between 6 and 12.
6, ____, ____, ____, 12
1
2
3
4
5
a n = a1 + (n − 1)d
a 5 = a1 + (5 − 1)d
12 = 6 + 4d
6 = 4d
6 3
d= =
4 2
Arithmetic Means:
6+
3 15
=
2
2
€
€
1. Find two arithmetic means between 3 and 15.
2. Find three arithmetic means between 3 and 13.
3. Find three arithmetic means between 6 and 11.
15 3 18
+ =
=9
2 2 2
€
9+
3 21
=
2
2
Algebra II
Unit 8
9
Geometric Sequences
A geometric sequence is a sequence in which the ratio, r, between successive terms is always the
same number.
Formula:
a n = a1 • r n−1
A geometric sequence: 4, 8, 16, 32, ... common ratio is 2.
Example 1: Find the eighth term of 4, 8, 16, 32, ...
a n = a1 • r n−1
a 8 = 4 • 28−1
a 8 = 4 • 27
a 8 = 4 • 128
a 8 = 512
1. Find the common ratio for each geometric sequence.
A. 3, 9, 27
B. 8, 2,
1 1
,
2 8
2. Find the missing term in each geometric sequence:
A. 2, 8, 32, ____, ...
B. ____, 3, 1,
1
, ...
3
Algebra II
Unit 8
10
3. The first term of a geometric sequence is 2. The common ratio is -2. Write the first four
terms.
4. The first term of a geometric sequence is
1
. The common ratio is 2. Write the first four
4
terms.
5. Find the 8th term of the geometric sequence: 1, 2, 4, ...
3
6. Find the 6th term of the geometric sequence: -6, -3, − , ...
2
7. A tank contains 8000 liters of water. Each day, one half of the remaining water is removed.
How much water will be in the tank on the eighth day?
8. Jane owns 20 shares of Contex, Inc. If the stock splits (she receives 2 shares for every share
she owns) every December for 5 years, how many shares will she own at the end of the fifth
year? (At the end of the first year she owns 40 shares.)
9. Find the 5th term of the geometric sequence:
2 1 3
, , ,...
3 2 8
10. Find the ninth term of the geometric sequence: 1, 3, 9, 27, ...
Algebra II
Unit 8
11
Introduction to Series and Sigma Notation
If you add the terms of a sequence, the result is called a series:
Sequence: 3, 5, 7, 9, 11,...
Series: 3+5+7+9+11+ ...
Sn =
the nth partial sum of a series
S1=
3
S2 =
3+5 = 8
S3 =
3+5+7 = 15
S4 =
3+5+7+9 = 24, and so forth.
Mathematicians invented a symbol telling us how to construct a partial sum:
Sigma: ∑
Let's see how to use sigma notation:
4
∑ (2) (3k ) = (2)(31) + (2)(32 ) + (2)(33 ) + (2)(34)
k=1
= 6 + 18 + 5 4 + 162
= 240
Evaluate the expression by writing the terms and adding them up:
1.
3.
4
∑ (4k − 3)
k=1
3
∑ (k + 6)
k=1
2.
4.
5
∑ (3k − 2)
k=1
4
∑ 5n
n=1
Algebra II
Unit 8
12
Sum of an Arithmetic Series
Example 1: Find the sum of the first fifteen terms of 2+5+8+...
a1 = 2
n = 15
d=3
Now we need to find the fifteenth term:
an = a1 + (n − 1)d
a15 = 2 + (15 − 1)(3)
a15 = 2 + (14)(3)
a15 = 2 + 42
a15 = 44
Using our new formula:
n(a1 + an )
2
15(2 + 44)
=
2
sn =
s15
S15 =
690
2
= 345
S15 =
S15
15(46)
2
Algebra II
Unit 8
13
10
∑ (7 − 2k)
Example 2:
k=1
a1 = 7 − 2(1) = 7 − 2 = 5
n=10
Sn =
a10 = 7 − 2(10) = 7 − 20 = −13
n(a1 + an )
2
10(5 + (−13))
2
10(−8)
=
2
S10 =
S10
S10 =
−80
= −40
2
n(a1 + an )
2
an = a1 + (n − 1)d
Sn =
1. Find the sum of the first sixteen terms of the arithmetic series: -7-5-3-...
2. Find the sum of the first ten terms of the arithmetic series: 7+12+17...
3. Find the sum of the first twenty terms of the arithmetic series: -13-11-9-...
Algebra II
4.
5.
Unit 8
100
∑
(3k + 5)
k=1
200
∑
n=1
−n
6. In a lecture hall, the front row has 18 seats. Each succeeding row has 4 more seats than the
row ahead of it. How many seats are there in the first 12 rows?
Sum of a Geometric Series
Example 1: Find the sum of the series 2+6+18+54+162.
a1 = 2
n=5
a1(1− r n )
sn =
1− r
s5 =
2(1−35 )
1− 3
S5 =
2(−242)
−2
S5 = 242
r=3
14
Algebra II
Unit 8
Example 2:
a1 = 2
Sn =
15
4 k
∑2
k=1
n=4
r=2
a1(1− r n )
1− r
2(1−24 )
S5 =
1− 2
S4 =
2(−15)
−1
S4 = 30
Sn =
a1(1− r n )
1− r
1. Find the sum of the first five terms of the geometric series: 4+8+16+...
2. Find the sum of the first five terms of the geometric series: 2-8+32-...
3. Find the sum of the geometric series: 1+4+16+64+256.
4.
4 k−1
∑3
k=1
Algebra II
Unit 8
11
5. ∑ 2k
n=1
6. How much would you have saved at the end of seven days if you had set aside $.64 on the
first day, $.96 on the second day, $1.44 on the third day and so on?
The Sum of an Infinite Geometric Series
This formula can be used to find the sum of an infinite geometric series when |r|<1.
Example 1: Find the sum of 5 −
a1 = 5
s=
a1
1− r
5
S=
1+
S=
1
3
5
4
3
S=
€
r=−
5 3 15
3
• =
=3
1 4 4
4
1
3
5 5 5
+ −
+ ...
3 9 27
16
Algebra II
Unit 8
17
Find the sum of each infinite geometric series:
1. 4+2+1+...
2. 12 + 4 +
3. 5 +
4
+ ...
3
5 5
+ + ...
3 9
Example: Drop a ball from 1 foot in the air. Suppose it consistently bounces up exactly
up the distance it fell. How far does the ball travel after the first bounce?
1
+
+
1st bounce
Start
after the first bounce
s∞ =
1 1 1 1 1 1 1
1
+ + + + + +
+
+ ...
2 2 4 4 8 8 16 16
 1
 1
 1
 1
s∞ = 2  + 2  + 2  + 2  + ...
 2
 4
 8
 16 
s∞ = 1+
s=
a1
1− r
1 1 1
+ + + ...
2 4 8
a1 = 1
r=
1
2
+ ...
1
way
2
Algebra II
Unit 8
S=
S=
18
1
1
1−
2
1
1
2
1 2
S= • =2
1 1
1. A ball is dropped from a height of 12 meters. Suppose it consistently bounces up exactly
1
3
of the way up the distance it fell. How far does the ball travel after the first bounce?
2. A ball is thrown vertically upward a distance of 54 meters. After hitting the ground, it
2
rebounds
the distance fallen and continues to rebound in the same manner. What distance
3
does the ball cover before coming to rest?
Algebra II
Unit 8
19
Formulas
1.
2.
3.
4.
5.
the nth term of an
Arithmetic Sequence
The nth term of an
Geometric Sequence
a n = a1 • r n−1
Sn =
Sn =
(
n a1 + a n
)
2


a11− r n 
1− r
a
S= 1
1− r
The Sum of the first n
terms of an Arithmetic
Series
The Sum of the first n
terms of a Geometric
Series
The Sum of an infinite
Geometric Series where
-1<r<1
Algebra II
Unit 8
20
Additional Problems: Please do your work on a separate piece of paper.
1. Find the first three terms of each sequence:
A. a n = 6n
B. a n = n + 2
3
C. a n = n +1
€
€
€
2. Find the next 3 terms of the sequence.
Find the rule for the nth term.
A. 4, 8, 12, 16, ...
B. –20, –26, –32 , –38, ...
3. For each of the following sequences, determine if it is arithmetic, geometric, or neither. If it is
arithmetic, find d. If it is geometric, find r.
A. 4, 7, 10, 13, .......
B. 3, 6, 12, 24, 48, ........
C. 2, 6, 24, 120, ......
D. 5,
5 5 5
, ,
,....
3 9 27
E. 1, 4, 9, 16, .........
4. Calculate a100 for the sequence: 17, 22, 27, 32, ...
5. Calculate a100 for the geometric sequence with first term a1 = 35 and common ratio r = 1.05.
€
6. The number 68 is a term in the arithmetic sequence with a1 = 5 and d = 3. Which term is it?
€
7. A geometric sequence has a1 = 17 and r = 2. If an = 34816, find n.
Algebra II
Unit 8
21
8. Find four arithmetic means between 55 and 85.
9. Find five arithmetic means between -91 and -67.
10. Find three arithmetic means between -257 and -397.
3
11. ∑ (2n + 3) =
n=1
4
12. ∑ n 2
n=1
th
13. Find the 127 partial sum of the arithmetic series with a1 = 17 and d = 4.
14. Find S34 for the geometric series with a1 = 7 and r = 1.03.
15. Given the sequence: 15, 9, 3, -3, ... Find the 50th term.
16. Given the sequence: 4, 6, 9, ... Find the 25th term.
17. The sixth term of an arithmetic sequence is -2 and the twelfth term is -14. find a1 and d.
18. The first term of an arithmetic sequence is 3 and the common difference is 2. Find the 900th
term.
19. Find the sum of the first 30 terms of a geometric series whose first term is 1 and whose
common ratio is 2.
20. Find the 14th term of the geometric sequence: 1, 2, 4, ...
3
21.
∑ (k + 2) =
k=1
22. Find the sum of the first 15 terms: 2+5+8+...
€
23. Find the sum of the first sixteen terms of the series: -7-5-3-...
24. Find the sum of the first ten terms of the series: 7+12+17+22+...
Algebra II
Unit 8
25. Find the sum of the first twenty terms of the series: -13-11-9-7-...........
26.
4
∑ 5n
n=1
27. Find the sum of the first 12 terms of the series: 1, 3, 9, 27...
100
28. Evaluate: ∑ (3k + 5)
k=1
200
29. Evaluate: ∑ −n
n=1
22