SEQUENCES
AND
SERIES
Algebra 2 & Trigonometry Lab
Miss Kersting
Room 206
Period 1
Name:____________________________
Day 1 – Sigma Notation
Summation (Sigma) Notation
7
Steps to Evaluate Sigma Notation:
Example:
 3i  2
i 4
Using the example: You are finding the sum of each term of the expression.
1.) The “i = 4” below  is the value of i for the first term. Plug this value in for i and
evaluate the expression.
2.) Continue plugging in consecutive values and evaluating each term until you reach the “7”
above the  .
3.) Add the sequence of numbers.
7
1.)
Evaluate
 3i  2
i 4
5
2.)
Evaluate
k
k 1
3
3.)
Evaluate
3
 
2 x x1
x 1
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Multiple Choice:
4.)
Which of the following represents the sum 3 + 6 + 12 + 24 + 48?
5
3
4
i
(1)
i 1
(2)
6
i 0
4
i 1
(3)
 32
48
i
i 0
(4)
i
i 3
5.)
Express the sum using sigma notation: 9 + 16 + 25 + … + 100
6.)
Express the sum using sigma notation: 6  8 10 12 14 16
7.)
Express the sum using sigma notation: 2  4  8  16  32
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Practice Problems:
Directions: Evaluate each expression indicated in sigma notation.
4
8.)
i
x
x0
4
9.)
x
2
x
x 1
3
10.)
3 2 x
x 0
11.)
Express the sum using sigma notation: 7  9  11  13  15  17
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Day 1 – Sigma Notation
HOMEWORK
**Complete all Practice Problems from class work that have not been completed.**
Directions: Evaluate each expression indicated in sigma notation.
8
1.)
2.)
 3 j  5
j 4
1 4
13  3k

2 k 1
5
3.)
3 n
n 2

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 2k  3 



k

k 1 
4
4.)
5.)
 2i 
6.)
Express the sum using sigma notation: 5 + 9 +13 + 17 + 21 + 25
5
2
i 1
Answers to Day 1 Homework
1.) 65
2.) 11
3.) 42
4.) 14.25
5.) 110
6.)
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6
5
x 1
x 0
 4 x  1 or   5  4 x 
Day 2 – Arithmetic Sequences/Geometric Sequences
Do Now: Questions 1 & 2
2.) Which represents the sum of 6+15+26+39?
5
1.) Which is equivalent to
3  j  4 ?
5
(1)
 3 j  4
j 2
(1)
 2n  2
5
(3)
 3 j  12
(2)
 n
j 2
 j 4



3 
j 2 
5
5
(2)
(4)
 4  j 
2
 10
n4
j 2
Sequence – a set of numbers written in a given order.
Steps to Find the Terms of a Sequence:
1.) Substitute into the given formula.
2.) Evaluate.
Directions: Find the first 3 terms of each sequence.
3.)
an  n  5
4.)
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(3)
n4
7
an  12  3n
 n
5
7
j 2

n
n 2
 3n 2 



n 2  2 
5
(4)
2

Arithmetic Sequences
COMMON DIFFERENCE
Geometric Sequences
COMMON RATIO
Formula: an  a1  n 1d
Formula:
an  a1r n 1
Steps to Determine the Common Difference:
1. Look at two consecutive terms.
2. Subtract the first term from the second term, and the second term from the third term.

3. If the difference is the same between each set, this is the common difference.
Directions: Find the common difference in each sequence.
5.) 5, 11, 17, 23, 29, …
6.) 20, 16, 12, 8, 4, …
7.) What is the common difference in the arithmetic sequence defined by the following formula?
t n  6n  3
Steps to Determine the Common Ratio:
Take two consecutive terms and divide the second term by the first term.
Directions: Find the common ratio in each sequence.
8.) 10, 30, 90, 270, 810, …
9.) 64, 32, 16, 8, 4, …
10.) What is the common ratio in the geometric sequence defined by the following formula?
n 1
tn  5  3
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Steps to Finding the nth Term of an Arithmetic or Geometric Sequence:
First, determine whether it is an arithmetic or geometric sequence by deciding whether you are
adding a constant to each term (arithmetic), or multiplying a constant to each term (geometric).
Arithmetic Sequence
Geometric Sequence
1.) Find the common difference.
2.) Use the formula an  a1  n 1d and
substitute the values into the formula.
3.) Simplify.
1.) Find the common ratio.
n 1
2.) Use the formula an  a1r and
substitute the values into the formula.
3.) Simplify.
11.) Find the 
20th term of the sequence:
-12, -7, -2, 3, …
12.) Find the 10th term of the sequence:
1 1 1
1, , , , ...
2 4 8
13.) Find the 35th term of the sequence:
106, 99, 92, 85, …
14.) Find the 12th term of the sequence:
3, 6, 12, 24, …
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Writing a Rule for the nth term of an Arithmetic Sequence:


1.) Substitute the first given term into the formula an  a1  n 1 d to create an equation.
2.) Substitute the second given term into the formula to create a second equation.
3.) Write the two equations as a system of equations and either subtract them or use
substitution. (Remember common difference.)
4.) Find d and substitute it into either equation to find a1.
the original formula to find the rule.
5.) Substitute the values for d and a1 into
15.) Write a rule for the nth term of an arithmetic sequence that has two given terms of
a4  36 and a9  61 .
Writing a Rule for the nth term of an Geometric Sequence:
n 1
1.) Substitute the larger n into the formula an  a1r
to create an equation.
2.) Substitute the smaller n into the formula to create a second equation.
3.) Divide the first equation by the second equation. (Remember common ratio)
4.) Find r and substitute it into either equation to find a1.
5.) Substitute the values for r and a1 into the original formula.
16.) Write a rule for the nth term of a geometric sequence that has two given terms of
a4  4 and a7  32 .
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Day 2 – Arithmetic Sequences/Geometric Sequences
HOMEWORK
Directions: Find the first 5 terms of each sequence.
n2
1.) a n  n 2
2.) a n 
n
Directions: Find the common difference or common ratio in each sequence.
3.) 6, 12, 18, 24, …
4.) 99, 33, 11, …
Directions: Find the requested term for each sequence.
5.) Find the 15th term of the sequence:
6.) Find the 8th term of the sequence:
-11, -8, -5, -2, …
2, 8, 32, 128, …
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7.) Given an arithmetic sequence, find d if
a2  6 and a36  30 .
8.) Given a geometric sequence, find r if a2  6
and a5  0.048 .
9.) Find the 25th term of the given arithmetic sequence if t3  4  5i, t6  13  11i .
Answers to Day 2 Homework
1.) 1, 4, 9, 16, 25
1
4.)
5 3 7
3
2.) 3, 2, , ,
3 2 5
5.) 31
3.) 6
6.) 32,768
7.)
12
17
1
5
9.) 70  49i
8.) 0.2 or
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Day 3 – Recursive Definition & Arithmetic Mean/Geometric Mean
Do Now: Questions 1 & 2
1.) Which of the following geometric sequences 2.) What is the common difference of the
has a common ratio of -2?
arithmetic sequence 6, 10, 14, 18?
1
(1) 24, -12, 6, -3, …
(3) , -1, 2, -4, …
10
2
(1)
(2) -4 (3) 4 (4) 10
6
(2) 1, -1, -3, -5, …
(4) -2, -4, -8, -16, …
Recursive Definition – allows any term of a sequence, except the first, to be computed from the
previous term.
Steps to Find Terms of a Recursive Sequence:
1.) Substitute a1 into the given formula and evaluate.
2.) Substitute the value from step 1 into the formula and evaluate.
3.) Continue to substitute the previous value into the formula until you find the number of
terms.
Directions: Find the first four terms of each of the recursive sequences.
a1  2
a1  4
3.)
4.)
a n  an 1  n
a n  2an 1  4
5.)
a1  6
a n  3an 1  1
6.)
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a1  4
a n  2an 1  10
Steps to Find a Certain Term using a Recursive Sequence:
1.) Determine if it is a recursive sequence by looking for a  n 1 .
2.) Plug in the given a n for a  n 1 and evaluate.
3.) Plug in the value found in the previous step into the given formula and continue until the
desired a n is found.
7.) Find the 7th term for the sequence where a3  12 and an  2an1 .
8.) Find the 4th term where a8  24 and an  2an1 .
9.) Find the 8th term where a4  4 and an  3an1  1 .
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Steps to Form a Sequence using Arithmetic Means:
1.) Add the two given numbers.
2.) Divide by 2. (We then have a1, ____,a3, _____,a5 )
3.) Find a 2 by adding a1 and a 3 , then divide the value by 2.
4.) Find a 4 by adding a 3 and a 5 , then divide the value by 2.
10.) Form a sequence that has three arithmetic means between 4 and 28.
Steps to Form a Sequence using Geometric Ratios:
n 1
1. Use the formula an  a1r
to find r with the given information.
2. Use the common ratio to determine the remaining numbers in the sequence.
11.) Form a sequence that has three geometric means between 4 and 324.
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Day 3 – Recursive Definition & Arithmetic Mean/Geometric Mean
HOMEWORK
Directions: Find the first five terms of each of the recursive sequences.
a1  4
a1  6
1.)
2.)
a n  an 1  5
a n  2an 1  n
3.)
a1  4
a n  3an 1  5
4.)
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a1  3
a n  2an 1  4
5.) Find the 5th term for the sequence where
a3  10 and an  2an1 .
6.) Find the 2nd term for the sequence where
a6  30 and an  2an1 .
7.) Form a sequence that has three arithmetic means between 16 and 632.
Answers to Day 3 Homework
1.) 4, 9, 14, 19, 24
2.) 6, 10, 17, 30, 55
3.) -4, -17, -56, -173, -524
4.) 3, -10, 16, -36, 68
5.) 40
15
or 1.875
8
7.) 16, 170, 324, 478, 632
6.)
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Day 4 – Arithmetic Series/Geometric Series
Do Now: Questions 1 & 2
1.) Find the first four terms of the sequence:
a1  3
2.) Find the 8th term for the sequence where
a3  4 and a n  2a n1  1 .
a n  an 1  5
Steps to Find the Sum of a Series for a Specified Number of Terms:
1.) Determine whether it is an arithmetic or geometric series by determining either the
common difference or common ratio.
2.) Use the formula on the formula sheet to determine the sum for the specified number of
terms.
Sum of a Finite Arithmetic Series:
n  a1  an 
Sn 
2
Sum of a Finite Geometric Series:
Sn 
a1 1  r n 
1 r
Directions: Find the sum of each of the following series.
3.) 3  6  9 12  ... for the first 33 terms
4.) 3  6 12  24... for the first 50 terms
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2 2
5.) 6  2    ... for the first 10 terms
3 9
6.) 4  7 10 13  ... for the first 100 terms
7.) Madison was determined to help clean her local park. She collected one bag of trash the first
week, 2 bags the second week, 3 bags the third week, and continued at the same rate.
Assuming she continues this process, how many bags of trash will she collect in 26 weeks?
If Madison collected one bag of trash the first week, 2 bags the second week, 4 bags the third
week, and continued at the same rate, how many bags of trash would she collect in 26 weeks?
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Day 4 – Arithmetic Series/Geometric Series
HOMEWORK
Directions: Find the sum of each of the following series.
1.) 10  20  30  40  ... for the first 20 terms
2.) 2  8  32  ... for the first seven terms
Answers to Day 4 Homework
1.) 2100
2.) 655
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