Algebra 2: Chapter 12 Notes Packet
Name:______________________________
**Included inside: Assignment Sheet, Notes, Study Guide, Homework Worksheets**
Algebra 2 Chapter 12 Assignment Sheet
*All assignments are due the day after they are assigned*
Date
33
W
1/12
In-Class
Homework
12.2: Arithmetic Sequences
34
Th
1/13
12.3: Geometric Sequences and Series
35
F
1/14
12.4: Sums of Infinite Geometric
Sequences
HW33:
Pg. 806 #4-20even
HW#33 Worksheet (pg 21 in this packet)
HW34:
Pg. 814 #4-20even
HW#34 Worksheet (pg 20 in this packet)
HW35:
Pg. 823 #8-22 even
HW#35 Worksheet (pg 19 in this packet)
**No School on Monday 1/17**
36
T
1/18
37
W
1/19
38
Th
1/20
12.1: Sums of Arithmetic Sequences
HW36:
Pg. 798 #2-26 even, 27, 38-58 even
HW#36 Worksheet (pg 18 in this packet)
Chapter 12 Test and Notes Check on
Thursday
HW37:
Chapter 12 review sheet (pgs. 15-17 in this
packet)
Chapter 12 Review
Chapter 12 Test and Notes Check on
Thursday
HW38:
Chapter 7 final review sheet
Chapter 8 final review sheet
Print: Final Review Packet
Chapter 12 Test and Notes Check
Practice Final on Tuesday, 1/25
Final Exam for periods 1, 2 on Thursday, 1/27
for periods 3, 4 on Friday, 1/28
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Chapter 12 Sequences and Series
Notes 33 : Section 12-2: Arithmetic Sequences
Definition of an Arithmetic Sequence
An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a
constant amount. The difference between consecutive terms is called the common difference, d, of the
sequence.
Formula for the General Term ( an ) of an Arithmetic Sequence
The nth term (the general term) of an arithmetic sequence with first term a1 and common difference d is:
an  a1   n  1 d
Practice: In 1-4, write the formula for the general term of the arithmetic sequence then find the
indicated term using the given information.
1. Find an and a16 when a1 = 9 and d = 2.
3. Find an and a5 when a7 = 34 and a18 = 122
2. Find an and a7 when d = −3 and a16 = 73
4. Find an and a6 when a2 = 8 and a20 = -136
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The Sum of the First n Terms of an Arithmetic Sequence
The sum, S n , of the first n terms of an arithmetic sequence is given by
n
Sn   a1  an 
2
th
In which a1 is the first term and an is the n term.
For Examples 5-7, find each sum.
5.) Find the sum of the first 20 terms of the arithmetic sequence: 4, 10, 16, 22,…..
6.) Find the sum of the first 60 positive event integers.
7.) Find the sum of the even integers between 21 and 45.
Sigma Notation:

i
20
8.) Evaluate:
 2i  4
i 1
82
9.) Evaluate:
 3n  5
n 1
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10.) Find n given S n = 175 for the arithmetic sequence: 4, 7, 10, 13, 16, 19,…….
11.) Find n given S n = 366 for the arithmetic sequence: 3, 8, 13, 18, 23,……
12.) Find n given S n = 308 for the arithmetic sequence: 3, 8, 13, 18, 23,……
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Exponential and Logarithmic Equation Review
*You must do these problems in order to get full credit for your notes, even if we do not do them in
class.
For #13-18, solve the exponential equation. Give the approximate value rounded to three
decimal places if necessary.
13. e x  18
14. 10 x  350
15. e2 x  42
16. 2 x  1  6
17. 42 x  16
18.
 
3 3x
2  1  10
8
In #19-24, Solve the logarithmic equation. Give the approximate value rounded to three decimal
places if necessary.
19. ln x  5
20. log x  2
21. log 2 x  1.5
22. 7ln x  21
23. 2  log 2 3x  8
24. log 2 ( x  4)  log 2 ( x  3)  3
HW #33: Pg. 806 #4-20even; HW#33 Worksheet (pg 21 in this packet)
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Notes 34: Section 12-3: Geometric Sequences and Series
Definition of a Geometric Sequence
A geometric sequence is a sequence in which each term after the first is obtained by multiplying the
preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the
common ratio of the sequence.
Formula for the General Term ( an ) of a Geometric Sequence
The nth term (the general term) of a geometric sequence with first term a1 and common ratio r is:
an  a1r n 1
1.) Write the first five terms of each geometric sequence given: a1 = 5 and r = 3.
For #2-4, write the rule for the nth term of the geometric sequence using the given information.
3. a2 = 45 and a5 = −1215
2. a1 = 4 and r = 2
4. a3 = 18 and a6 =
2
3
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The Sum of the First n Terms of a Geometric Sequence
The sum, S n , of the first n terms of a geometric sequence is given by
Sn 

a1 1  r n

1 r
In which a1 is the first term and r is the common ratio (r ≠ 1).
Examples: For #5-8, find the indicated sum for the geometric sequence.
10
5. Find the sum of the first 10 terms of: 1, 5, 25, 125, 625,…….
6.
i 1
 6  2
i 1
12
7.
i 1
1
 4  2 
i 1
9
8.)
i 1
 5  3
i 1
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Examples: For #9-10, find the number of terms n need to get the given sum S n .
1023
9. S n = 3906 given 1, 5, 25, 125, 625,……
10. S n 
given 4, 2, 1, ……
128
Exponential and Logarithmic Equation Review
*You must do these problems in order to get full credit for your notes, even if we do not do them in
class.
For #11-16, solve the exponential equation. Give the approximate value rounded to three decimal
places if necessary.
11. log 1 27  x
12. log 2 3x  log 2 6  4
13. log 4 5  log4 x  log4 60
3
14. 3e x  4  13
15. 20.1x  6  12
16. 5 x  27
HW #34: Pg. 814 #4-20even; HW#34 Worksheet (pg 20 in this packet)
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Notes #35: Section 12-4: Finding Sums of Infinite Geometric Series
The Sum of an Infinite Geometric Series
The sum of an infinite geometric series with first term a1 and common ratio r is given by
S
a1
1 r
provided r  1 . If r  1 , the series has no infinite sum.
Examples: For #1-4, find the sum of each infinite geometric series.

i 1
1 1 1
 .....
1.) 1   
3 9 27
1
2.)  4  
2
i 1  
3.) 4 + 8 + 16 + ….
1 1
4.) 16,  4, 1,  ,
4 16
5.) Find the common ratio of an infinite
4
geometric series with first term a1 =  and has
3
a sum of −2.
6.) Find the common ratio of an infinite
geometric series with first term a1 = 28 and has a
sum of 35.
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Sequence and Series Review:
7.) Find the 25th term of the sequence:
1, -2, -5, -8, …
8.) Find the 16th term of the sequence:
-108, 36, -12, 4, …
9.) Given an arithmetic sequence, if a1  19 and
10.) Evaluate:
a7  49 , find a23
5
 (3i  2)
i 1
11.) Given the arithmetic sequence such that
a2  9 and a7  37 , find a35
12.) Given an arithmetic sequence such that
a2 = 7 and a20 = 205, find an and a6 .
13.) Given the geometric sequence such that
1
a2  16 and a7   , find S15
2
14.) Given an arithmetic sequence such that
a1  5 and d = 3, find n given Sn  1274
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Exponential and Logarithmic Equation Review
15.) Graph f ( x)  3x  4  6 . Graph the parent function then
transform the graph using h and k values. Then state the
domain, range and the equation of asymptote for the function.
16.) Write log 2 7 with the common base.
17.) Solve: log3  2 x 1  2
For #18-19, use loga 5  0.516 and loga 7  1.213 to approximate the value of each expression.
25
18.) log a 35
19.) log a
7
HW #35: Pg. 823 #8-22 even; HW#35 Worksheet (pg 19 in this packet)
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Notes #36: Section 12-1: Sequences and Summation Notation
After completing section 12-1 you should be able to do the following:
1. Find the terms of a sequence given the rule.
2. Write the rule for a given sequence.
3. Use summation notation.
4. Memorize the formulas to find sums of special series and use them to find the sum
Definition of a Sequence
A sequence is a function whose domain is a set of consecutive integers. If the domain is not specified,
it is understood that the domain starts with 1. The values in the range are called terms of the sequence.
Domain:
1
2
3
4……. n
The relative position of each term
Range:
a1
a2
a3
a4 .… an
Terms of the sequence
A finite sequence has a limited number of terms. An infinite sequence continues without stopping.
Finite Sequence: 2, 4, 6, 8
Infinite Sequence: 2, 4, 6, 8, ……
A sequence can be specified by an equation, or rule. For example both sequences above can be
described by the rule:
an = 2n
Examples: For #1-2, write the first four terms of each sequence whose rule is given.
2n
n
2
1. an   1  n 
2. an 
n4
Finding the rule of a sequence given its terms.
In order to write a rule you have to pay attention to its domain also know as the term number, n. The
term number is its position in the sequence. The symbol used to show the value of the term is an , so
the first term is a1 the second term is a2 and so on. In order to write the rule it is all about describing
the patterns you see between the term number and the value of the term. It is really about guess and
check at this point. So let’s try some.
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Examples: For #3-4, write the rule, an , for the given sequence.
3.) 1, 6, 11, 16,…..
4.) 1.2, 4.2, 9.2, 16.2,…..
5.)
2 2 2 2
, , ,
,......
3 6 9 12
6.) –4, 8, –12, 16,……
Summation Notation
It is sometimes useful to find the sum of the first n terms of a sequence. A compact notation of
expressing the sum of the first n terms is called summation notation.
Summation Notation
The sum of the first n terms of a sequence ins represented by the summation notation
n
 ai  a1  a2  a3  ......  an 1  an ,
i 1
where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of
summation.
Examples: For #7-8, expand and evaluate the sum.
4
7.
 2i
i 1
6
8.
 2  h  1 !
h 3
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Writing summation notation
In order to write a series in summation notation you must find the rule ( an ). Once you have found the
rule that relates the term number to its value need to figure out the number of terms in the sequence
given. If it goes on without end we state the limit of summation is ∞.
Examples: For #9-10, write the series using summation notation.
9.) 7 + 10 + 13 + 16 + 19
10.)
1 2 3 4 5 6 7
     
4 5 6 7 8 9 10
Special Formulas to find sums of series without having to add up each individual term of the
series:
Sum of n terms of a
Sum of first n positive
Sum of squares of first n
constant c:
integers:
positive integers:
n
 c  cn
i 1
n  n  1
i  2
i 1
n
n
 i2 
i 1
n  n  1 2n  1
6
Only use these formulas when n (upper limit of summation is a large number)
Examples: For #11-13, find the sum of the series.
35
11.)
 10
k 1
40
12.)
 4n
n 1
20
13.)
 w2
w 1
HW #36: Pg. 798 #2-26 even, 27, 38-58 even; HW#36 Worksheet (pg 18 in this packet)
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HW #37--Chapter 12 Review Sheet
Name: _____________________________
For question 1-3, write a rule for the nth term of the arithmetic sequence. Then find a20 .
9 11 13
1.) 1, 7, 13, 19, 25, …
2.) ,5, , 6, ,…
3.) -4, -2, 0, 2, 4,…
2
2
2
For questions 4-6, write a rule for the nth term of the arithmetic sequence.
4.) d = -4, a1 =7
5.) a4  18, a10  48
6.) a7  22, a11  34
For questions 7-8, find (a) the sum of the first n terms of the arithmetic series and (b) find n for
the given sum, S n .
5
7
7.) 2   3   4  ...
(a) n = 50
(b) S n =1005
2
2
8.) 32, 24, 16, 8, 0, …
(a) n = 30
(b) S n = -880
For questions 9-10, write a rule for the nth term of the geometric sequence.
1
9.) a3  18, a6  486
10.) a8  , a15  243
9
For questions 11-12, find (a) the sum of the first n terms of the geometric series and (b) find n for
the given sum, S n .
11.) 1  (4)  16  (64)  ...
(a) n = 8
(b) S n =  819
12.)
9
2 4
 1    ...
2
9 81
(a) n = 20
(b) S n =5.5
(more on next page!!)
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For questions 13-15, find the sum of the series.
11
i
1
13.)  7  
2
i 0  
36
14.)
 (5  3i)
i 1

2
15.)  4  
3
n 0  
n
For questions 16-18, find the sum of the infinite geometric series if it has one.

1
16.)  2  
4
n 1  
n 1

3
17.)  4  
2
n 0  
n

 1
18.)  10   
2
n 1 
n 1
For questions 19-20, find the common ratio of the infinite geometric series with the given sum
and first term.
16
19.) S  4, a1  3
20.) S  , a1  8
3
For questions 21-22, find the first five terms of each sequence.
n
21.) an  2n2
22.) an 
n2
In 23-25, evaluate.
50
25
23.)
 3h 2
h 1
24.)
 2y
y 1
30
25.)
 (t 2  2t  5)
t 1
(more on next page!!)
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In 26-34, solve each equation. (a) Give the exact value (before plugging anything into your
calculator). (b) Round to three decimal places, if needed.
26.) 24 x  5  163 x  2
27.) e53 x  4  6
28.) log(2 x  1)  3
29.) 9log5 x  4  11
30.) ln(4 x)  6  8
31.) 5  2ln x  5
32.) log x 64  2
33.) log x 54  3
34.) log33 (2 x  1)  log33 ( x  2)  1
(now you’re done!  )
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HW#36 Worksheet
Name: ______________________
For questions #1-3, write a rule for the nth term of the arithmetic sequence. Then find a20 .
7
9 11
1.) 3, 7, 11, 15, 19, …
2.) , 4, ,5, ,…
3.) -3, -1, 1, 3, 5,…
2
2
2
For questions #4-6, write a rule for the nth term of the arithmetic sequence.
4.) d = -6, a1 =7
5.) a4  18, a10  54
6.) a7  22, a11  42
For questions #7-8, find (a) the sum of the first n terms of the arithmetic series and (b) find n for
the given sum, S n .
5
7
7.) 2   3   4  ...
(a) n = 40
(b) S n =135
2
2
8.) 32, 24, 16, 8, 0, …
(a) n = 50
(b) S n = -5544
In #9-17, solve the final exam review problems. Leave your answers in (a) exact form and (b) in
approximate form, rounded to four decimal places.
11. log 2 x  log 2 ( x  2)  3
9. 45 x 1  8 x 1
10. log3  5x  7   2
12. log x 34  2
13. 3log6 x  5  12
14. 4  3ln x  2
15. e3 2 x  5  9
16. ln(3x)  5  11
17. 5e2 x  1  19
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HW#35 Worksheet
Name: ______________________
For #1-8, complete each problem about sequences and series.
1. Write a rule for the arithmetic sequence:
2. Write a rule for the geometric sequence:
-3, 5, 13, 21, …
1
,  2, 8,  32,...
2
3. Find n for the sequence 4, -3, -10, -17, … if
Sn  884
4. Find n for the sequence -2, 6, -18, 54, … if
Sn  3281
5. Find a13 and S13 for the sequence:
97, 75, 53, 31…
6. Find a20 of the geometric sequence for whom
1
a3  and a6  1
8
13
7. Evaluate:
1
8. Evaluate:  3  
i 1  3 
 (5i  8)
7
i 1
i 1
In #9-17, solve the final exam review problems. Leave your answers in (a) exact form and (b) in
approximate form, rounded to four decimal places.
9. 32 x  3  4
10. 2 x  1  6
11. e  x  6  1
12. 3e x  18
15. 5  2ln x  5
13.
3 3x
 2   1  10
8
16. 7  log x  4
14. log 2 x  1.5
17. ln 4x  6  8
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HW#34 Worksheet
Geometric Sequences
Name: _____________________
Formulas you need to memorize for Geometric Sequences:
a1 (1  r n )
n 1
General Rule: an  a1 (r )
Sum: Sn 
1 r
For 1-4, find the value of ‘n’ for the given sum S n for each problem.
1. 1  4  16  64  ..., Sn  341
2. 7  (21)  63  (189)  ..., Sn  3829
4. 90  30  (10) 
3. 1  9  81  729  ...., Sn  820
10
 ..., Sn  66.67
3
Find the rule for the geometric sequence describe by the given information.
5. a4 = 108; r = 3
6. a5 = 20; a9 = 320
7. a2 = 4, a4 = 5
8. a2 = -24, a5 = 1536
In #9-17, solve the final exam review problems. Leave your answers in (a) exact form and (b) in
approximate form, rounded to four decimal places.
9. 10 x  350
10. e2 x  5  12
11. 2 x  7  10
12. e4 x  3  7
15. 4  ln x  1
13.
2 2x
e  12
3
16. 2  log3 2 x  3
14. 2 log x  10
17. ln(2 x  3)  ln(2 x  1)
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HW#33 Worksheet:
Arithmetic Sequences
Name: ____________________
Formulas you need to memorize for Arithmetic Sequences:
n
2
Sum: s   a1  an 
General Rule: an  a1   n 1 d
In #1-6, Write a rule for the nth term of the arithmetic sequence with the given information.
1. d = 14, a14 = 46
2. a2 = −28, a20 = = 52
4. a5 = 17, a17 = 77
5. d =
5
, a8
3
= 24
3. a1 = −2, a9 =
1
6
6. d = 4.1, a16 = 48.2
In #7-10, Find n for the given sum Sn.
7. 3 + 8 + 13 + 18 + 23 +……; Sn = 366
8. 50 + 42 + 34 + 26 + 18 +…..; Sn = 182
9. 2 + 9 + 16 + 23 + 30 + …..; Sn = 1661
10. 2 + 16 + 30 + 44 + 58 + ….; Sn = 2178
In #11-22, solve the final exam review problems. Leave your answers in (a) exact form and (b)
in approximate form, rounded to four decimal places.
11. e x  18
12. e3 x  6  10
13. 2e x  10
14. 42 x  3  1
15. 4e 2 x  3  5
16. 7ln x  21
17. 3log x  1  13
18. 3  ln x  5
19. log 2 5 x  1
20. 2  log 2 3x  8
21. ln(5 x  1)  ln(3x  2)
22. ln(4 x  1)  2
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