12.1 Arithmetic Sequences & Series 1. A sequence is an ordered list of numbers. Each number in the list is called a term of the sequence. The first term of a sequence is denoted as 1 a . The second term is denoted as a2. The term in the n th position is called the n th term and is denoted as n a . The term before n a is 1 n a . A sequence is a function whose range is the terms of the sequence and the domain is the position of each term. 2. Finding the Next Term: To find the next term in an arithmetic sequence, first find the common difference, then add the common difference to the next term. 3. Find the next three terms. a. . . . , 10 , 1 , 12 b. 7, 10, 13, … c. r – 4, r – 1, r + 2 4. Arithmetic sequences: The sequence 1 a , 2 a , 3 a , 4 a , …. n a is arithmetic if there is a number d such that: 2 a – 1 a = d 3 a – 2 a = d Where d is the common difference. Ex: 6, 9, 12, 15, ….3n + 3 The common difference is 3 because 9 – 6 = 3. d = 3 Ex: 2, -3, -8, -13, …, -5n + 7 The common difference is –5 because -3–2 = -5. d = -5 5. Explicit Formula: a formula that defines the n th term. We will look at c as being the 0 a (a sub not) term. Think y-intercept! The book will use d n a a n 1 1 or d a a n n 1 (when recursive) 6. Recursive Sequence: is a sequence in which each term is defined using the previous terms. Each Arithmetic Sequence can be written recursively using d a a n n 1 Ex: Find a formula of an arithmetic sequence whose common difference is 4 and whose first term is 3. n a = dn + c We know d = 4. 1 a = 3. So 0 a = 3 – 4. 0 a = -1 n a = 4n – 1. The terms of this sequence are: 3, 7, 11, 15, …, 4n – 1. Ex: Find the formula of the arithmetic sequence whose first term is 3 and whose second term is –1. n a = dn + c We know 1 a = 3 and 2 a = -1. So d = -4. 0 a must be 3–(-4) = 7 n a = -4n + 7 The terms of this sequence are: 3, -1, -5, -9, …, -4n + 7. an = dn + c
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12.1 Arithmetic Sequences & Series
1. A sequence is an ordered list of numbers. Each number in the list is called a term of the sequence. The
first term of a sequence is denoted as 1a . The second term is denoted as a2. The term in the nth position
is called the nth term and is denoted as na . The term before na is 1na .
A sequence is a function whose range is the terms of the sequence and the domain
is the position of each term.
2. Finding the Next Term: To find the next term in an arithmetic sequence, first find the common
difference, then add the common difference to the next term.
3. Find the next three terms.
a. ...,10,1,12 b. 7, 10, 13, … c. r – 4, r – 1, r + 2
4. Arithmetic sequences:
The sequence 1a , 2a , 3a , 4a , …. na is arithmetic if there is a number d such that:
2a – 1a = d
3a – 2a = d Where d is the common difference.
Ex: 6, 9, 12, 15, ….3n + 3 The common difference is 3 because 9 – 6 = 3. d = 3
Ex: 2, -3, -8, -13, …, -5n + 7 The common difference is –5 because -3–2 = -5. d = -5
5. Explicit Formula: a formula that defines the nth term.
We will look at c as being the 0a (a sub not) term. Think y-intercept!
The book will use dnaan 11 or daa nn 1 (when recursive)
6. Recursive Sequence: is a sequence in which each term is defined using the previous terms.
Each Arithmetic Sequence can be written recursively using daa nn 1
Ex: Find a formula of an arithmetic sequence whose common difference is 4 and whose first term is 3.
na = dn + c We know d = 4. 1a = 3. So 0a = 3 – 4. 0a = -1
na = 4n – 1. The terms of this sequence are: 3, 7, 11, 15, …, 4n – 1.
Ex: Find the formula of the arithmetic sequence whose first term is 3 and whose second term is –1.
na = dn + c We know 1a = 3 and 2a = -1. So d = -4. 0a must be 3–(-4) = 7
na = -4n + 7 The terms of this sequence are: 3, -1, -5, -9, …, -4n + 7.
an = dn + c
7. Find the 35th term in the sequence 11, 4, -3, …
8. Find the 20th term in the sequence for which 1a = -27 and d = 3.
9. Find the first term in the sequence for which 4a = 229 and d = 8.
10. The fifth term of an arithmetic sequence is 25 and the 12th term is 60.
Write the first several terms of this sequence.
5a = 25 12a = 60
12a = 5a + 7d (where 7 is the difference in the term numbers).
60 = 25 + 7d 35 = 7d d = 5
Since 5a = 25 we can subtract 5 to get each term in the sequence down to the first.
5, 10, 15, 20, 25
11. Arithmetic means: the terms between any two nonconsecutive terms of an arithmetic sequence.
The terms between 2 given terms of an arithmetic sequence are called arithmetic means.
10, 13, 16, 19, 22 10, 14, 18, 22
3 arithmetic means 2 arithmetic means
12. Form an arithmetic sequence that has five arithmetic means between -11 and 19.
13. Form an arithmetic sequence that has six arithmetic means between -12 and 23.
14. Summation Notation: the sum of a sequence is also known as an Arithmetic Series.
m
m
k
k ccccc
...321
1
15. Sigma Notation: the sum of the first n terms of a sequence (called a series)
Ex:
5
2
2i
i Ex:
8
3
)32(k
k Ex:
10
5
)3(j
j
16. The Sum of an Arithmetic Series: Sn = naan
12
This means that we add the first and last terms,
then multiply by the number of terms divided by 2.
Ex: Find the sum of the integers from 1 to 500.
Sn = naan
12
n = 500, a1 = 1 and an = 500 Sn = 50012
500 = 250(501) = 125, 250
17. Find the sum of the first 27 terms in the series -14, -8, -2, ……+ 142.
18. Find the sum of the first 32 terms in the series -12, -6, 0, ….
19. Find n for a series for which 1a = 5, d = 3, and 440nS .
20. Nimisha starts a college savings account for her daughter on her sixth birthday. She plans to deposit $25
the first month and then increase the deposit by $5 each month. How much will she have deposited in
twelve years?
21. The number of seats in the first row is 20, the second row is 23, the third row is 26, and so on. How
many seats are in Row 16? How many seats is there altogether in those 16 rows?
DAY 1 HW
12.1 pg. 660-661 #5-37 odd, 41, 43
Find the next five terms in each arithmetic sequence.
5. 5, 9, 13, … 7. 5, -1, -7, … 9. 1.5, 3, 4.5, …
11. –n, 0, n, … 13. b, -b, -3b,…
15. Find the 79th term in the sequence -7, -4, -1, ….
17. Form an arithmetic sequence that has one arithmetic mean between 12 and 21.
Solve. Assume that each sequence is an arithmetic sequence.
19. Find the 19th term in the sequence for which 1a = 11 and d = -2.
21. Find n for the sequence for which na = 37, 1a = -13, and d = 5.
23. Find the first term in the sequence for which d = -2 and 7a = 3.
25. Find d for the sequence for which 1a = 4 and 11a = 64.
27. Find the sixth term in the sequence ...,31,32
29. Find the 43rd term in the sequence -19, -15, -11, …
31. Form a sequence that has one arithmetic mean between 36 and 48.
33. Form a sequence that has two arithmetic means between 2 and 10.
35. Find the sum of the first 11 terms in the series – 1 + 1 + 3 + …
37. Find n for a series for which 1a = -7, d = 1.5, and nS = -14.
41. Terri works after school at the Find Foods Supermarket. One day, Terri had to stack cans of soup in a
grocery display in the form of a triangle. On the top row, there was only one can. Each row below it
contained one more can that the one above it. On the bottom row, there were 21 cans. If all the cans
were the same size, how many cans were in the display?
43. Michael is a chocoholic. On New Year’s Day, he ate one piece of chocolate. On the next day, he ate 2
pieces. On each subsequent day, he ate one additional piece of candy.
a. How many pieces of candy did he eat on the last day of January?
b. How many pieces of did he eat during the month of January?
Day 2 Sequences and Series Notes
1. Recursive Formula:
A formula for a sequence that gives the value of a term in terms of the preceding term . The
first term is represented by , the second term in represented by , the third term in represented by
, and so forth.
Explicit Formula: or if Arithmetic Sequence!
2. Find the next three terms in each sequence.
a. 80, 77, 74, 71, 68, … b. 4, 8, 16, 32, 64, … *c. 0, 3, 7, 12, 18, … * d. ...32
1,
16
1,
8
1,
4
1,
2
1
3. Now write the recursive formula for the sequences above.
4. If 221 a and 31 nn aa , find the next three terms.
5. If 641 t and 12
1 nn tt , find the next four terms.
6. If 31 a , 52 a and 12 4 nnn aaa , find the third, fourth and fifth terms..
7. Find the first four terms.
a. ntn 316 b. 143 kak c. 2
11
nnan
daa nn 1 daa nn 1
8. If the domain values are 5,3,0,1 , find the corresponding range values for 52 ntn .
10. Sigma Notation
Simplifies the process of writing out the sum of a series
10
2
2n
n
11.
5
1
2k
k 12.
2
1
)2(k
k
13.
3
0
2k
k 14.
5
2
2k
k
(no parentheses, careful)
15.
3
1
2
n
n
5
2n
n 16.
9
5
)]1(43[k
n
(typo: must be k’s)
is read as the sum of n2 as n increases
from 2 to 10
last value of n
first value of n
Homework: Worksheet #1 and Worksheet #2
DAY 2 HW
Worksheet 1
I. Give the first four terms of each sequence:
1. 3,5 11 nn ttt 2. nttt nn 11 ,10 3. 11 2,3 nn ttt
4. 12,4 11 nn ttt
II. Give the third, fourth, and fifth terms of each sequence: