Name _____________________________
Date ______________
Period _____
Algebra 2
Chapter 12 Packet
Part 1: Introduction to Sequences
The following are sequences . . .
A:
2, 4, 6, 8, 10, . . .
B:
1, 4, 9, 16, 25
C:
3, 9, 15, 21 . . .
D:
A sequence is: __________________________________________
__________________________________________
__________________________________________
__________________________________________
Which of the sequences above are finite? ________
Which are infinite? _________
What differentiates a finite sequence from an infinite sequence? _____________________________
term: refers to the position of the number in the sequence, some sequences start with term a 1.
In sequence A what is a4? ______
In sequence A, could you figure out a10? _____
In sequence B what is a1? ______
Write the first six terms of the following sequences. Start with n = 1, then find a20.
1. an = n2 + 1
2. an = (-2)n - 1
Go back to the sequences from before. Describe the pattern and write a rule for the nth term
3.
2, 4, 6, 8, 10, . . .
4.
5.
3, 9, 15, 21 . . .
6.
1, 4, 9, 16, 25
7. Find the first four terms of the sequence a n = n(5n + 5).
a) 10, 30, 60, 100
b) 5, 15, 20, 25
c) 5, 25, 50, 85
d) 10, 30, 65, 110
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8. The number of lilies a large nursery can sell each day after April 1 is modeled by a sequence whose general term is
an = 1700 – 75n, where n is the number of days after April 1. Find the number of lilies that can be sold on
April 6th, 7th , and 8th.
a) 1325, 1250, 1175
b) 1175, 1250, 1175
c) 1250, 1175, 1100
d) 1400, 1325, 1250
9. Write three different sequences that begin 1, 3, … Explain the pattern in each sequence.
Part 2: Arithmetic Sequences (a specific type of sequence)
Arithmetic sequence: a sequence that you add the same number to get the value of the next term.
common difference: the value that you add to get the next term. This value may be positive or negative or 0.
Each of the following are arithmetic sequences, find the next four terms and the value of d.
1. 5, 8, 11, …
2. -4, -6, -8, …
3. 100, 93, 86, …
5.
7
2
, 6,
17
2
4. -24, -19, -14, …
,11, ...
6. 4.8, 4.1, 3.4, …
7. a1 = 7, d = 7
8. a1 = -8, d = ½
Graph the terms for the following arithmetic sequences:
A:
-4, -1, 2, 5, 8, . . .
B:
-10, -8, -6, -4, . . .
d=
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
d=
12
11
10
9
8
7
6
5
4
3
2
1
1
-12, -2, 8, 18, . . .
d=
12
11
10
9
8
7
6
5
4
3
2
1
-1
C:
2
3
4
5
6
7
8
9
-1
10 -1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
12
11
10
9
8
7
6
5
4
3
2
1
1
2
2
3
4
5
6
7
8
9 -1
10
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
1
2
3
4
5
6
7
8
* What does the graph of an arithmetic sequence look like? ______________
* What is the relationship between the slope of the line and the common difference? ______________
Therefore to write a general rule for an arithmetic sequence we can use
an = d(n) + b
where (n, an) is a term given
d is the common difference and
b (which is a0) is the term we have to find!!
Examples: If the sequence is arithmetic, there is a general rule. Write the rule for the nth term of the sequence
1. 4, 9, 14, 19, . . .
2. ¼, 0, - ¼, - ½ , - ¾
3. One term of an arithmetic sequence is a 19 = 48.
The common difference is d = 3
4. One term of an arithmetic sequence is a5 = 33.
The common difference is d = 5
5. Two terms of an arithmetic sequence are
a8 = 21 and a27 = 97.
6. Two terms of an arithmetic sequence are.
a6 = 13 and a14 = 25
7. a7 = 21, a13 = 42
8. d = 14, a6 = -10
9.
10.
-12, -5, 2, 9, …
3
45, 31, 17, 3, …
Part 3 - Geometric Sequences
What makes a sequence geometric??
 the ratio of any term to the previous term is constant (the common ratio r)
 multiply/divide by the same value to get the next term
The following are geometric sequences. Write the next two terms for each sequence and the r value.
1. 2, 4, 8, 16, 32
2. 625, 125, 25, 5, 1, . . .
3. 1, - 4, 16, - 64, 256, . . .
Find the first six terms of the geometric sequence given
1. a2 = 6, r = 3
2. a1 = 10, r =
5
2
MUST MEMORIZE:
If the sequence is geometric, there is a rule for the nth term
an  a1 (r )n 1
where a1 is the first term and r is the common ratio
Examples:
Write the rule for the nth term of the sequence. Then find a15
1. 4, 20, 100, 500, . . .
2. 152, -76, 38, -19, . . .
3. -6, -2, 
2
2
,  ,…
3
9
4. r =
5. One term of a geometric sequence is a4 = 32.
The common ratio is r = 2
1
, a2 = 4
10
6. One term of a geometric sequence is a6 = -96.
The common ratio is r = -2
4
7. Two terms of a geometric sequence are
a3 = 24 and a5 = 96.
8. Two terms of a geometric sequence are
a2 = -12 and a4 = -3.
9. You want to earn some money, so you ask your parents if they will pay you money to keep your room clean every
day for the month of June. You explain to your parents that you are not asking for much money, just a penny the
first day, 2 pennies the second day, double that to 4 cents the next day, then 8 cents on the 4th day on so on. I'm
sure your parents would love to see your room clean!
If they say yes, how much would they pay you on June 10th, how much on June 30th?
Part 4 - Arithmetic/Geometric Series
Below are three examples of a series, what do you notice to be the difference between a sequence and a series???
Series A: 1+4+7+10+13
Series B:
1 1 1
1
1
1
  
 
 ...
3 8 15 24 35 48
A series . . .
 Is when the terms of a sequence are added together
 Can be infinite or finite
In a series, we can write out a list of terms which we will add together or we can use summation notation.
Write in words the correct way to read
4
 n  2 ? ________________________________________________
n 1
What is the index of summation? __________
What is the general term or rule? _______________
What is the lower limit of summation? _______
What is the upper limit of summation? _______
What is the sum? _________________________
n2 =
4
n 1
IN YOUR CALCULATOR!!!
Set MODE to seq
Go to sum (2nd STAT→ MATH #5sum)
Go to seq (2nd STAT→ OPS #5seq)
Enter in general rule, index of summation, lower limit of summation, upper limit of summation
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Evaluate the following sums using your calculator
2
Series E:  4  
n 0  3 
10
Series D:
5
 50  2i
i 1
4
Series F:
8
3
Series G:
i
2
n
2
i 5
n1
Example: A health club rolls its towels and stacks them in layers on a shelf. Each layer of towels has one less towel
than the layer below it.
a) If there are 20 towels on the first row, how many towels are on the 6th row?
Do you want a series or a sequence?
b) How many towels are there if the stack has 12 rows? Do you want a series or a sequence?
Part 5 - Review 12.1-12.4
Checklist:
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Do you know what the difference is between a sequence and a series?
Do you know how to tell whether a sequence (or series) is geometric, arithmetic, or neither?
Do you know the formulas for both arithmetic and geometric sequences?
Can you find the general rule given a list of terms?
Can you find the general rule for both arithmetic and geometric given 2 terms?
Can you find the arithmetic rule given the common difference and 1 term?
Can you find the geometric rule given the common ratio and 1 term?
Do you know how to use your calculator to find the sum of a series?
Do you know how to rewrite a series using summation notation?
Do you know how to tell if an infinite geometric series has a sum or not?
Do you know the formula for finding the sum of an infinite geometric series?
In all problems use the correct math notation and show work!!!!
1. Tell whether the sequence is arithmetic, geometric, or neither. Explain your answer.
a) 80, 77, 74, 71, 68,...
b) 0, 4,9,15, 22,...
c) 2, 0.5,1, 2.5,...
d) 6, 4, ,
8 16
,...
3 9
6
2. Write the first six terms of the sequence starting with
3. If an 
n  1 . an 
2n 1
(n  1)2
2n
, what is a10 ?
2n
4. Write the next term in the sequence, and then write the rule for the nth term.
a) 1,1, 1,1, 1,...
b) 4, 6,8,10,12.......
c) 12, 6,3, 1.5,...
d)
e) 0,3,8,15,...
f) 3, 6, 12, 24, 48...
5. Write the rule for an arithmetic sequence if
1 2 3 4
, , , ,...
2 3 4 5
a11  31 and a18  10 .
6. Write the rule for a geometric sequence if the common ratio is 2 and
7
a14  2048 .
7. Write the rule for a geometric sequence if
a5  162 and a8  4374 .
8. Find the sum of each series.
a)
 1 
b) 12 

 3 
n 1
25
5
 8  2n
n 15
9. Use the formula Sum 

11
a)   
n 1 4  3 
a1
1r
n 1
n2  1
c) 
n 1 2n
100
to evaluate the following:
n
b) 1 - 3 + 9 - 27 + 81 - 243 . . . .
9. Write the first six terms of the sequence of even numbers.
a) Write the rule for this sequence.
b) What is the 150th term in this sequence?
c) What is the sum of the first 200 terms?
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