SlideShare a Scribd company logo

Class_Powerpoint_Sequences_Arithmetic_and_Geometric_with_Series_Finite_and_Infinite.ppt

Download as PPT, PDF
M
MargieCDeSagun

1. The document discusses arithmetic and geometric sequences. 2. Arithmetic sequences are defined by adding a common difference to get the next term, while geometric sequences multiply by a common ratio. 3. Examples are provided of finding terms in arithmetic and geometric sequences using formulas.

Education
1 of 50
Download to read offline
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
12.1 – Arithmetic Sequences and Series
An introduction………… 1, 4, 7,10,13 9,1, 7, 15 6.2, 6.6, 7, 7.4 , 3, 6        Arithmetic Sequences ADD To get next term 2, 4, 8,16, 32 9, 3,1, 1/3 1,1/ 4,1/16,1/ 64 , 2.5 , 6.25      Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms 35 12 27.2 3 9    Geometric Series Sum of Terms 62 20/3 85/ 64 9.75
USING AND WRITING SEQUENCES The numbers in sequences are called terms. You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1.
The domain gives the relative position of each term. 1 2 3 4 5 DOMAIN: 3 6 9 12 15 RANGE: The range gives the terms of the sequence This is a finite sequence having the rule an = 3n, where an represents the nth term of the sequence. USING AND WRITING SEQUENCES n an
Writing Terms of Sequences Write the first six terms of the sequence an = 2n + 3. SOLUTION a1 = 2(1) + 3 = 5 1st term 2nd term 3rd term 4th term 6th term a2 = 2(2) + 3 = 7 a3 = 2(3) + 3 = 9 a4 = 2(4) + 3 = 11 a5 = 2(5) + 3 = 13 a6 = 2(6) + 3 = 15 5th term
Writing Terms of Sequences Write the first six terms of the sequence f(n) = (–2)n – 1 . SOLUTION f(1) = (–2) 1 – 1 = 1 1st term 2nd term 3rd term 4th term 6th term f(2) = (–2) 2 – 1 = –2 f(3) = (–2) 3 – 1 = 4 f(4) = (–2) 4 – 1 = – 8 f(5) = (–2) 5 – 1 = 16 f(6) = (–2) 6 – 1 = – 32 5th term
Arithmetic Sequences How do I define an arithmetic sequence and how do I use the formula to find different terms of the sequence?
Arithmetic Sequences and Series Arithmetic Sequence: sequence whose consecutive terms have a common difference. Example: 3, 5, 7, 9, 11, 13, ... The terms have a common difference of 2. The common difference is the number d. To find the common difference you use an+1 – an Example: Is the sequence arithmetic? –45, –30, –15, 0, 15, 30 Yes, the common difference is 15 How do you find any term in this sequence? To find any term in an arithmetic sequence, use the formula an = a1 + (n – 1)d where d is the common difference.
Find the 14th term of the arithmetic sequence 4, 7, 10, 13,…… 1 ( 1) n a a n d    14 a  (14 1)   4 3 4 (13)3   4 39   43 
1 ( 1) n a a n d    In the arithmetic sequence 4,7,10,13,…, which term has a value of 301? 301 4 ( 1)3 n    301 4 3 3 n    301 1 3n   300 3n  100 n 
Find the next four terms of –9, -2, 5, … Arithmetic Sequence 2 9 5 2 7        7 is referred to as the common difference (d) Common Difference (d) – what we ADD to get next term Next four terms……12, 19, 26, 33
Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of x, 2x, 3x, … Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -32k
Vocabulary of Sequences (Universal) 1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference      n 1 n 1 n nth term of arithmetic sequence sum of n terms of arithmetic sequen a a n 1 d n S a a 2 ce       
Given an arithmetic sequence with 15 1 a 38 and d 3, find a .    1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  x 15 38 NA -3   n 1 a a n 1 d       38 x 1 1 5 3     X = 80
63 Find S of 19, 13, 7,...    1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  -19 63 ?? x 6   n 1 a a n 1 d       ?? 19 6 1 ?? 353 3 6      353   n 1 n n S a a 2     63 63 3 3 S 2 19 5    63 1 1 S 052 
16 1 Find a if a 1.5 and d 0.5   Try this one: 1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  1.5 16 x NA 0.5   n 1 a a n 1 d      16 1.5 0. a 16 5 1    16 a 9 
n 1 Findnif a 633, a 9, and d 24    1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  9 x 633 NA 24   n 1 a a n 1 d      633 9 2 1 x 4    633 9 2 24 4x    X = 27
1 29 Find dif a 6 and a 20    1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  -6 29 20 NA x   n 1 a a n 1 d      1 20 6 29 x     26 28x  13 x 14 
Find two arithmetic means between –4 and 5 -4, ____, ____, 5 1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  -4 4 5 NA x   n 1 a a n 1 d       1 5 4 4 x     x 3  The two arithmetic means are –1 and 2, since –4, -1, 2, 5 forms an arithmetic sequence
Find three arithmetic means between 1 and 4 1, ____, ____, ____, 4 1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  1 5 4 NA x   n 1 a a n 1 d       4 1 x 1 5    3 x 4  The three arithmetic means are 7/4, 10/4, and 13/4 since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence
Find n for the series in which 1 n a 5, d 3,S 440    1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  5 x y 440 3   n 1 a a n 1 d      n 1 n n S a a 2     y 5 3 1 x      x 40 y 4 2 5       1 2 x 440 5 5 x 3       x 7 x 440 2 3     880 x 7 3x   2 0 3x 7x 880    X = 16 Graph on positive window
12.2 – Geometric Sequences and Series
1, 4, 7,10,13 9,1, 7, 15 6.2, 6.6, 7, 7.4 , 3, 6        Arithmetic Sequences ADD To get next term 2, 4, 8,16, 32 9, 3,1, 1/3 1,1/ 4,1/16,1/ 64 , 2.5 , 6.25      Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms 35 12 27.2 3 9    Geometric Series Sum of Terms 62 20/3 85/ 64 9.75
Vocabulary of Sequences (Universal) 1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio    n 1 n 1 n 1 n nth term of geometric sequence sum of n terms of geometric sequ a a r a r 1 S r 1 ence           
Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 – 2 vs. 9/2 – 3… not arithmetic 3 9/ 2 3 1.5 geometric r 2 3 2      3 3 3 3 3 3 2 2 2 9 2, 3, , , , 2 9 9 9 2 2 2 2 2 2       9 2, 3, , , 27 81 243 4 8 , 2 16
1 9 1 2 If a ,r , find a . 2 3   1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio  1/2 x 9 NA 2/3 n 1 n 1 a a r   9 1 1 2 x 2 3            8 8 2 x 2 3   7 8 2 3  128 6561 
Find two geometric means between –2 and 54 -2, ____, ____, 54 1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio  -2 54 4 NA x n 1 n 1 a a r      1 4 54 2 x    3 27 x   3 x   The two geometric means are 6 and -18, since –2, 6, -18, 54 forms an geometric sequence
2 4 1 2 Find a a if a 3 and r 3     -3, ____, ____, ____ 2 Since r ... 3  4 8 3, 2, , 3 9     2 4 8 10 a a 2 9 9             
9 Find a of 2, 2, 2 2,... 1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio  x 9 NA 2 2 2 2 r 2 2 2    n 1 n 1 a a r     9 1 x 2 2     8 x 2 2  x 16 2 
5 2 If a 32 2 andr 2, find a    ____, , ____,____ ____ ,32 2 1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio  x 5 NA 32 2 2  n 1 n 1 a a r     5 1 32 2 x 2      4 32 2 x 2   32 2 x 4  8 2 x 
*** Insert one geometric mean between ¼ and 4*** *** denotes trick question 1 ,____,4 4 1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio  1/4 3 NA 4 x n 1 n 1 a a r   3 1 1 4 4 r   2 r 1 4 4   2 16 r   4 r    1 ,1, 4 4 1 , 1, 4 4 
7 1 1 1 Find S of ... 2 4 8    1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio  1/2 7 x NA 1 1 1 8 4 r 1 1 2 2 4      n 1 n a r 1 S r 1        7 1 1 2 2 x 1 2 1 1                          7 1 1 2 2 1 2 1                          63 64 
Section 12.3 – Infinite Series
1, 4, 7, 10, 13, …. Infinite Arithmetic No Sum 3, 7, 11, …, 51 Finite Arithmetic   n 1 n n S a a 2   1, 2, 4, …, 64 Finite Geometric   n 1 n a r 1 S r 1    1, 2, 4, 8, … Infinite Geometric r > 1 r < -1 No Sum 1 1 1 3,1, , , ... 3 9 27 Infinite Geometric -1 < r <1 1 a S 1 r  
Find the sum, if possible: 1 1 1 1 ... 2 4 8     1 1 1 2 4 r 1 1 2 2    1 r 1 Yes      1 a 1 S 2 1 1 r 1 2     
Find the sum, if possible: 2 2 8 16 2 ...    8 16 2 r 2 2 8 2 2    1 r 1 No      NO SUM
Find the sum, if possible: 2 1 1 1 ... 3 3 6 12     1 1 1 3 6 r 2 1 2 3 3    1 r 1 Yes      1 2 a 4 3 S 1 1 r 3 1 2     
Find the sum, if possible: 2 4 8 ... 7 7 7    4 8 7 7 r 2 2 4 7 7    1 r 1 No      NO SUM
Find the sum, if possible: 5 10 5 ... 2    5 5 1 2 r 10 5 2    1 r 1 Yes      1 a 10 S 20 1 1 r 1 2     
The Bouncing Ball Problem – Version A A ball is dropped from a height of 50 feet. It rebounds 4/5 of it’s height, and continues this pattern until it stops. How far does the ball travel? 50 40 32 32/5 40 32 32/5 40 S 45 50 4 1 0 1 5 5 4     
The Bouncing Ball Problem – Version B A ball is thrown 100 feet into the air. It rebounds 3/4 of it’s height, and continues this pattern until it stops. How far does the ball travel? 100 75 225/4 100 75 225/4 10 S 80 100 4 4 3 1 0 1 0 3     
Sigma Notation
B n n A a   UPPER BOUND (NUMBER) LOWER BOUND (NUMBER) SIGMA (SUM OF TERMS) NTH TERM (SEQUENCE)
  j 4 1 j 2      2 1     2 2     3 2     2 4   18    7 4 a 2a       4 2      2 5      2 6      7 2  44    n n 0 4 0.5 2      0 0.5 2     1 0.5 2     2 0.5 2     3 0.5 2     4 0.5 2   33.5 
0 n b 3 6 5           0 3 6 5       1 3 6 5        2 3 6 5        ...  1 a S 1 r   6 15 3 1 5      2 x 3 7 2x 1                    2 1 2 8 1 2 9 1 ... 7 2 1 23              n 1 n 2 n 1 S a a 15 2 3 2 7 47       527 
  1 b 9 4 4b 3                    4 3 4 5 3 4 6 3 ... 4 4 3 19              n 1 n 1 n 1 S a a 19 2 9 2 4 79       784 
Rewrite using sigma notation: 3 + 6 + 9 + 12 Arithmetic, d= 3   n 1 a a n 1 d      n a 3 n 1 3    n a 3n  4 1 n 3n  
Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1 Geometric, r = ½ n 1 n 1 a a r   n 1 n 1 a 16 2         n 1 n 5 1 1 16 2         
Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4 Not Arithmetic, Not Geometric n 1 n a 20 2    n 1 n 5 1 20 2     19 + 18 + 16 + 12 + 4 -1 -2 -4 -8
Rewrite the following using sigma notation: 3 9 27 ... 5 10 15    Numerator is geometric, r = 3 Denominator is arithmetic d= 5 NUMERATOR:   n 1 n 3 9 27 ... a 3 3       DENOMINATOR:   n n 5 10 15 ... a 5 n 1 5 a 5n          SIGMA NOTATION:   1 1 n n 5n 3 3    

More Related Content

PPT
Arithmetic and geometric_sequences
Dreams4school
 
PDF
Geometric Sequence and Series.pdf
VnDr
 
PPT
Arithmetic Sequences and Series-Boger.ppt
JosephMuez2
 
PPT
Arithmetic Sequences and Series-Boger.ppt
GIDEONPAUL13
 
PPT
Arithmetic Sequences and Series-Boger.ppt
Angelle Pantig
 
PPT
Sequences_Arithmetic_Harmonic_Geometric.ppt
srei8348
 
PPTX
SEQUENCES AND SERIES
University of Johannesburg
 
PPT
Arithmetic Sequences and Series.ppt grade 10
JaneOlesco1
 
Arithmetic and geometric_sequences
Dreams4school
 
Geometric Sequence and Series.pdf
VnDr
 
Arithmetic Sequences and Series-Boger.ppt
JosephMuez2
 
Arithmetic Sequences and Series-Boger.ppt
GIDEONPAUL13
 
Arithmetic Sequences and Series-Boger.ppt
Angelle Pantig
 
Sequences_Arithmetic_Harmonic_Geometric.ppt
srei8348
 
SEQUENCES AND SERIES
University of Johannesburg
 
Arithmetic Sequences and Series.ppt grade 10
JaneOlesco1
 

Similar to Class_Powerpoint_Sequences_Arithmetic_and_Geometric_with_Series_Finite_and_Infinite.ppt (20)

PPT
Arithmetic Sequences and Series for Grade 10
JaneOlesco1
 
PPTX
Sequence.pptx
Ces Joanne Fajarito
 
PPT
Arithmetic Sequences and Series.ppt
John Marvin Canaria
 
PPT
Maths project work - Arithmetic Sequences
S.L.B.S Engineering College
 
PPT
425732902-Lesson-9-Sequences-and-Series-ppt.ppt
JohnJericPioquid1
 
PPTX
Chapter 1 arithmetic &amp; geometric sequence
Najla Nizam
 
PPT
Barisan dan deret .ingg
Fendik Bagoez
 
PPT
Barisan dan deret .ingg
Fendik Bagoez
 
PPTX
Generating patterns in Sequences_MATH10.pptx
libunaodesserie
 
PPTX
Patterns, sequences and series
Vukile Xhego
 
PPT
10.1-10.8.js-lecturenotes for students.ppt
samuel obara, Ph.D.
 
PDF
Arithmetic Sequence and Arithmetic Series
Joey Fontanilla Valdriz
 
PPTX
P2-Chp3-SequencesAndSeries from pure maths 2.pptx
ArafathAliMathsTeach
 
PPT
Geometric Sequences and Series it is for
SonikaNagi2
 
DOC
Sequence and series
jin lau
 
PPTX
P2-Chp3-Pure mathSequencesAndSeries.pptx
jonammarhaba
 
PPTX
Sequences, Series, and the Binomial Theorem
Ver Louie Gautani
 
PPT
Arithmetic sequences (1).ppt
DeepaIyer32
 
PPTX
Series in Discrete Structure || Computer Science
MubasharGhazi
 
PPT
Chapter 1 sequences and series
wan suriyani che wan ahmad
 
Arithmetic Sequences and Series for Grade 10
JaneOlesco1
 
Sequence.pptx
Ces Joanne Fajarito
 
Arithmetic Sequences and Series.ppt
John Marvin Canaria
 
Maths project work - Arithmetic Sequences
S.L.B.S Engineering College
 
425732902-Lesson-9-Sequences-and-Series-ppt.ppt
JohnJericPioquid1
 
Chapter 1 arithmetic &amp; geometric sequence
Najla Nizam
 
Barisan dan deret .ingg
Fendik Bagoez
 
Barisan dan deret .ingg
Fendik Bagoez
 
Generating patterns in Sequences_MATH10.pptx
libunaodesserie
 
Patterns, sequences and series
Vukile Xhego
 
10.1-10.8.js-lecturenotes for students.ppt
samuel obara, Ph.D.
 
Arithmetic Sequence and Arithmetic Series
Joey Fontanilla Valdriz
 
P2-Chp3-SequencesAndSeries from pure maths 2.pptx
ArafathAliMathsTeach
 
Geometric Sequences and Series it is for
SonikaNagi2
 
Sequence and series
jin lau
 
P2-Chp3-Pure mathSequencesAndSeries.pptx
jonammarhaba
 
Sequences, Series, and the Binomial Theorem
Ver Louie Gautani
 
Arithmetic sequences (1).ppt
DeepaIyer32
 
Series in Discrete Structure || Computer Science
MubasharGhazi
 
Chapter 1 sequences and series
wan suriyani che wan ahmad
 
Ad

Recently uploaded (20)

PDF
Pre independence Education in Inndia.pdf
goofy5603
 
PPTX
Pharmacology of Heart Failure /Pharmacotherapy of CHF
Rajshri Ghogare
 
PDF
Sports Quiz easy sports quiz sports quiz
nikunj2757
 
PPTX
Lesson notes of climatology university.
sgladness67
 
PDF
O5-L3 Freight Transport Ops (International) V1.pdf
labyankof
 
PDF
2.FourierTransform-ShortQuestionswithAnswers.pdf
Raji Murugan
 
PPTX
Final Presentation General Medicine 03-08-2024.pptx
ZaheerAhmad228692
 
PPTX
Cell Types and Its function , kingdom of life
ClaireMangundayao1
 
PDF
Module 4: Burden of Disease Tutorial Slides S2 2025
Jonathan Hallett
 
PDF
Saundersa Comprehensive Review for the NCLEX-RN Examination.pdf
sreejithcareers
 
PPTX
PPT- ENG7_QUARTER1_LESSON1_WEEK1. IMAGERY -DESCRIPTIONS pptx.pptx
KMDee
 
PPTX
GDM (1) (1).pptx small presentation for students
shrawanbpkihs
 
PDF
The Lost Whites of Pakistan by Jahanzaib Mughal.pdf
jahanzaibmughal6
 
PDF
3rd Neelam Sanjeevareddy Memorial Lecture.pdf
Academy of Grassroots Studies and Research of India (AGRASRI)
 
PDF
STATICS OF THE RIGID BODIES Hibbelers.pdf
pascualasturiaskaye4
 
PDF
Complications of Minimal Access Surgery at WLH
mohitsuren827
 
PDF
Insiders guide to clinical Medicine.pdf
AbhishekPatil315128
 
PPTX
human mycosis Human fungal infections are called human mycosis..pptx
shilpa939953
 
PDF
VCE English Exam - Section C Student Revision Booklet
jpinnuck
 
PDF
ANTIBIOTICS.pptx.pdf………………… xxxxxxxxxxxxx
HakHe
 
Pre independence Education in Inndia.pdf
goofy5603
 
Pharmacology of Heart Failure /Pharmacotherapy of CHF
Rajshri Ghogare
 
Sports Quiz easy sports quiz sports quiz
nikunj2757
 
Lesson notes of climatology university.
sgladness67
 
O5-L3 Freight Transport Ops (International) V1.pdf
labyankof
 
2.FourierTransform-ShortQuestionswithAnswers.pdf
Raji Murugan
 
Final Presentation General Medicine 03-08-2024.pptx
ZaheerAhmad228692
 
Cell Types and Its function , kingdom of life
ClaireMangundayao1
 
Module 4: Burden of Disease Tutorial Slides S2 2025
Jonathan Hallett
 
Saundersa Comprehensive Review for the NCLEX-RN Examination.pdf
sreejithcareers
 
PPT- ENG7_QUARTER1_LESSON1_WEEK1. IMAGERY -DESCRIPTIONS pptx.pptx
KMDee
 
GDM (1) (1).pptx small presentation for students
shrawanbpkihs
 
The Lost Whites of Pakistan by Jahanzaib Mughal.pdf
jahanzaibmughal6
 
3rd Neelam Sanjeevareddy Memorial Lecture.pdf
Academy of Grassroots Studies and Research of India (AGRASRI)
 
STATICS OF THE RIGID BODIES Hibbelers.pdf
pascualasturiaskaye4
 
Complications of Minimal Access Surgery at WLH
mohitsuren827
 
Insiders guide to clinical Medicine.pdf
AbhishekPatil315128
 
human mycosis Human fungal infections are called human mycosis..pptx
shilpa939953
 
VCE English Exam - Section C Student Revision Booklet
jpinnuck
 
ANTIBIOTICS.pptx.pdf………………… xxxxxxxxxxxxx
HakHe
 
Ad

Class_Powerpoint_Sequences_Arithmetic_and_Geometric_with_Series_Finite_and_Infinite.ppt

  • 1. 12.1 – Arithmetic Sequences and Series
  • 2. An introduction………… 1, 4, 7,10,13 9,1, 7, 15 6.2, 6.6, 7, 7.4 , 3, 6        Arithmetic Sequences ADD To get next term 2, 4, 8,16, 32 9, 3,1, 1/3 1,1/ 4,1/16,1/ 64 , 2.5 , 6.25      Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms 35 12 27.2 3 9    Geometric Series Sum of Terms 62 20/3 85/ 64 9.75
  • 3. USING AND WRITING SEQUENCES The numbers in sequences are called terms. You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1.
  • 4. The domain gives the relative position of each term. 1 2 3 4 5 DOMAIN: 3 6 9 12 15 RANGE: The range gives the terms of the sequence This is a finite sequence having the rule an = 3n, where an represents the nth term of the sequence. USING AND WRITING SEQUENCES n an
  • 5. Writing Terms of Sequences Write the first six terms of the sequence an = 2n + 3. SOLUTION a1 = 2(1) + 3 = 5 1st term 2nd term 3rd term 4th term 6th term a2 = 2(2) + 3 = 7 a3 = 2(3) + 3 = 9 a4 = 2(4) + 3 = 11 a5 = 2(5) + 3 = 13 a6 = 2(6) + 3 = 15 5th term
  • 6. Writing Terms of Sequences Write the first six terms of the sequence f(n) = (–2)n – 1 . SOLUTION f(1) = (–2) 1 – 1 = 1 1st term 2nd term 3rd term 4th term 6th term f(2) = (–2) 2 – 1 = –2 f(3) = (–2) 3 – 1 = 4 f(4) = (–2) 4 – 1 = – 8 f(5) = (–2) 5 – 1 = 16 f(6) = (–2) 6 – 1 = – 32 5th term
  • 7. Arithmetic Sequences How do I define an arithmetic sequence and how do I use the formula to find different terms of the sequence?
  • 8. Arithmetic Sequences and Series Arithmetic Sequence: sequence whose consecutive terms have a common difference. Example: 3, 5, 7, 9, 11, 13, ... The terms have a common difference of 2. The common difference is the number d. To find the common difference you use an+1 – an Example: Is the sequence arithmetic? –45, –30, –15, 0, 15, 30 Yes, the common difference is 15 How do you find any term in this sequence? To find any term in an arithmetic sequence, use the formula an = a1 + (n – 1)d where d is the common difference.
  • 9. Find the 14th term of the arithmetic sequence 4, 7, 10, 13,…… 1 ( 1) n a a n d    14 a  (14 1)   4 3 4 (13)3   4 39   43 
  • 10. 1 ( 1) n a a n d    In the arithmetic sequence 4,7,10,13,…, which term has a value of 301? 301 4 ( 1)3 n    301 4 3 3 n    301 1 3n   300 3n  100 n 
  • 11. Find the next four terms of –9, -2, 5, … Arithmetic Sequence 2 9 5 2 7        7 is referred to as the common difference (d) Common Difference (d) – what we ADD to get next term Next four terms……12, 19, 26, 33
  • 12. Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of x, 2x, 3x, … Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -32k
  • 13. Vocabulary of Sequences (Universal) 1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference      n 1 n 1 n nth term of arithmetic sequence sum of n terms of arithmetic sequen a a n 1 d n S a a 2 ce       
  • 14. Given an arithmetic sequence with 15 1 a 38 and d 3, find a .    1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  x 15 38 NA -3   n 1 a a n 1 d       38 x 1 1 5 3     X = 80
  • 15. 63 Find S of 19, 13, 7,...    1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  -19 63 ?? x 6   n 1 a a n 1 d       ?? 19 6 1 ?? 353 3 6      353   n 1 n n S a a 2     63 63 3 3 S 2 19 5    63 1 1 S 052 
  • 16. 16 1 Find a if a 1.5 and d 0.5   Try this one: 1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  1.5 16 x NA 0.5   n 1 a a n 1 d      16 1.5 0. a 16 5 1    16 a 9 
  • 17. n 1 Findnif a 633, a 9, and d 24    1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  9 x 633 NA 24   n 1 a a n 1 d      633 9 2 1 x 4    633 9 2 24 4x    X = 27
  • 18. 1 29 Find dif a 6 and a 20    1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  -6 29 20 NA x   n 1 a a n 1 d      1 20 6 29 x     26 28x  13 x 14 
  • 19. Find two arithmetic means between –4 and 5 -4, ____, ____, 5 1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  -4 4 5 NA x   n 1 a a n 1 d       1 5 4 4 x     x 3  The two arithmetic means are –1 and 2, since –4, -1, 2, 5 forms an arithmetic sequence
  • 20. Find three arithmetic means between 1 and 4 1, ____, ____, ____, 4 1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  1 5 4 NA x   n 1 a a n 1 d       4 1 x 1 5    3 x 4  The three arithmetic means are 7/4, 10/4, and 13/4 since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence
  • 21. Find n for the series in which 1 n a 5, d 3,S 440    1 a First term  n a nth term  n S sum of n terms  n number of terms  d common difference  5 x y 440 3   n 1 a a n 1 d      n 1 n n S a a 2     y 5 3 1 x      x 40 y 4 2 5       1 2 x 440 5 5 x 3       x 7 x 440 2 3     880 x 7 3x   2 0 3x 7x 880    X = 16 Graph on positive window
  • 22. 12.2 – Geometric Sequences and Series
  • 23. 1, 4, 7,10,13 9,1, 7, 15 6.2, 6.6, 7, 7.4 , 3, 6        Arithmetic Sequences ADD To get next term 2, 4, 8,16, 32 9, 3,1, 1/3 1,1/ 4,1/16,1/ 64 , 2.5 , 6.25      Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms 35 12 27.2 3 9    Geometric Series Sum of Terms 62 20/3 85/ 64 9.75
  • 24. Vocabulary of Sequences (Universal) 1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio    n 1 n 1 n 1 n nth term of geometric sequence sum of n terms of geometric sequ a a r a r 1 S r 1 ence           
  • 25. Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 – 2 vs. 9/2 – 3… not arithmetic 3 9/ 2 3 1.5 geometric r 2 3 2      3 3 3 3 3 3 2 2 2 9 2, 3, , , , 2 9 9 9 2 2 2 2 2 2       9 2, 3, , , 27 81 243 4 8 , 2 16
  • 26. 1 9 1 2 If a ,r , find a . 2 3   1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio  1/2 x 9 NA 2/3 n 1 n 1 a a r   9 1 1 2 x 2 3            8 8 2 x 2 3   7 8 2 3  128 6561 
  • 27. Find two geometric means between –2 and 54 -2, ____, ____, 54 1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio  -2 54 4 NA x n 1 n 1 a a r      1 4 54 2 x    3 27 x   3 x   The two geometric means are 6 and -18, since –2, 6, -18, 54 forms an geometric sequence
  • 28. 2 4 1 2 Find a a if a 3 and r 3     -3, ____, ____, ____ 2 Since r ... 3  4 8 3, 2, , 3 9     2 4 8 10 a a 2 9 9             
  • 29. 9 Find a of 2, 2, 2 2,... 1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio  x 9 NA 2 2 2 2 r 2 2 2    n 1 n 1 a a r     9 1 x 2 2     8 x 2 2  x 16 2 
  • 30. 5 2 If a 32 2 andr 2, find a    ____, , ____,____ ____ ,32 2 1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio  x 5 NA 32 2 2  n 1 n 1 a a r     5 1 32 2 x 2      4 32 2 x 2   32 2 x 4  8 2 x 
  • 31. *** Insert one geometric mean between ¼ and 4*** *** denotes trick question 1 ,____,4 4 1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio  1/4 3 NA 4 x n 1 n 1 a a r   3 1 1 4 4 r   2 r 1 4 4   2 16 r   4 r    1 ,1, 4 4 1 , 1, 4 4 
  • 32. 7 1 1 1 Find S of ... 2 4 8    1 a First term  n a nth term  n S sum of n terms  n number of terms  r commonratio  1/2 7 x NA 1 1 1 8 4 r 1 1 2 2 4      n 1 n a r 1 S r 1        7 1 1 2 2 x 1 2 1 1                          7 1 1 2 2 1 2 1                          63 64 
  • 33. Section 12.3 – Infinite Series
  • 34. 1, 4, 7, 10, 13, …. Infinite Arithmetic No Sum 3, 7, 11, …, 51 Finite Arithmetic   n 1 n n S a a 2   1, 2, 4, …, 64 Finite Geometric   n 1 n a r 1 S r 1    1, 2, 4, 8, … Infinite Geometric r > 1 r < -1 No Sum 1 1 1 3,1, , , ... 3 9 27 Infinite Geometric -1 < r <1 1 a S 1 r  
  • 35. Find the sum, if possible: 1 1 1 1 ... 2 4 8     1 1 1 2 4 r 1 1 2 2    1 r 1 Yes      1 a 1 S 2 1 1 r 1 2     
  • 36. Find the sum, if possible: 2 2 8 16 2 ...    8 16 2 r 2 2 8 2 2    1 r 1 No      NO SUM
  • 37. Find the sum, if possible: 2 1 1 1 ... 3 3 6 12     1 1 1 3 6 r 2 1 2 3 3    1 r 1 Yes      1 2 a 4 3 S 1 1 r 3 1 2     
  • 38. Find the sum, if possible: 2 4 8 ... 7 7 7    4 8 7 7 r 2 2 4 7 7    1 r 1 No      NO SUM
  • 39. Find the sum, if possible: 5 10 5 ... 2    5 5 1 2 r 10 5 2    1 r 1 Yes      1 a 10 S 20 1 1 r 1 2     
  • 40. The Bouncing Ball Problem – Version A A ball is dropped from a height of 50 feet. It rebounds 4/5 of it’s height, and continues this pattern until it stops. How far does the ball travel? 50 40 32 32/5 40 32 32/5 40 S 45 50 4 1 0 1 5 5 4     
  • 41. The Bouncing Ball Problem – Version B A ball is thrown 100 feet into the air. It rebounds 3/4 of it’s height, and continues this pattern until it stops. How far does the ball travel? 100 75 225/4 100 75 225/4 10 S 80 100 4 4 3 1 0 1 0 3     
  • 43. B n n A a   UPPER BOUND (NUMBER) LOWER BOUND (NUMBER) SIGMA (SUM OF TERMS) NTH TERM (SEQUENCE)
  • 44.   j 4 1 j 2      2 1     2 2     3 2     2 4   18    7 4 a 2a       4 2      2 5      2 6      7 2  44    n n 0 4 0.5 2      0 0.5 2     1 0.5 2     2 0.5 2     3 0.5 2     4 0.5 2   33.5 
  • 45. 0 n b 3 6 5           0 3 6 5       1 3 6 5        2 3 6 5        ...  1 a S 1 r   6 15 3 1 5      2 x 3 7 2x 1                    2 1 2 8 1 2 9 1 ... 7 2 1 23              n 1 n 2 n 1 S a a 15 2 3 2 7 47       527 
  • 46.   1 b 9 4 4b 3                    4 3 4 5 3 4 6 3 ... 4 4 3 19              n 1 n 1 n 1 S a a 19 2 9 2 4 79       784 
  • 47. Rewrite using sigma notation: 3 + 6 + 9 + 12 Arithmetic, d= 3   n 1 a a n 1 d      n a 3 n 1 3    n a 3n  4 1 n 3n  
  • 48. Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1 Geometric, r = ½ n 1 n 1 a a r   n 1 n 1 a 16 2         n 1 n 5 1 1 16 2         
  • 49. Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4 Not Arithmetic, Not Geometric n 1 n a 20 2    n 1 n 5 1 20 2     19 + 18 + 16 + 12 + 4 -1 -2 -4 -8
  • 50. Rewrite the following using sigma notation: 3 9 27 ... 5 10 15    Numerator is geometric, r = 3 Denominator is arithmetic d= 5 NUMERATOR:   n 1 n 3 9 27 ... a 3 3       DENOMINATOR:   n n 5 10 15 ... a 5 n 1 5 a 5n          SIGMA NOTATION:   1 1 n n 5n 3 3    