Math 34: Fall Arithmetic Sequences
ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesArithmetic SequencesSupplemental Material Not Found in You TextReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceMath 34: Fall 2016FormulaExamplesPartial SumsFormulaExamplesDo NOT print these slides!!There are printer friendly files on the website.HomeworkSeptember 9, 2016
Arithmetic SequencesArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework1 Arithmetic SequencesReviewReal World Examples2 General Way to Write an Arithmetic SequenceFormulaExamples3 Partial SumsFormulaExamples4 Homework
Arithmetic SequencesArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkRecall an Arithmetic Sequence is a sequence where thedifference between any two consecutive numbers in thesequence is constant.In other words: ak 1 ak d where d is a constant.
Arithmetic SequencesArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkRecall an Arithmetic Sequence is a sequence where thedifference between any two consecutive numbers in thesequence is constant.In other words: ak 1 ak d where d is a constant.Which of the following are Arithmetic Sequences?1231, 4, 7, 10, 13, . . .2, 4, 8, 16, 32, . . . 3, 7, 17, 27, . . .
Real World ExamplesArithmeticSequencesMath 34: Fall20161You start a new job and you’re told you salary is 29, 000for the first year, and that you’ll get a 1700 raise eachyear. What will your salary be in the third year? What willyour salary be in 10 years?2A new company has a loss of 2, 500 in its first month,but they expect their monthly profit to increase by 400each month. What is their profit in the 12th month?What is their total profit/loss of the year?ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework
Real World ExamplesArithmeticSequencesMath 34: Fall20161You start a new job and you’re told you salary is 29, 000for the first year, and that you’ll get a 1700 raise eachyear. What will your salary be in the third year? What willyour salary be in 10 years?2A new company has a loss of 2, 500 in its first month,but they expect their monthly profit to increase by 400each month. What is their profit in the 12th month?What is their total profit/loss of the year?ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkBoth these scenarios can be modeled by ArithmeticSequences, and we will develop tools to help us answerthese questions.
General Way to Write an Arithmetic SequenceArithmeticSequencesMath 34: Fall2016Consider the Arithmetic Sequence below. Notice the firstterm is 5 and the common difference is 2:ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework5, 7, 9, 11, 13, . . .
General Way to Write an Arithmetic SequenceArithmeticSequencesMath 34: Fall2016Consider the Arithmetic Sequence below. Notice the firstterm is 5 and the common difference is 2:ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework5, 7, 9, 11, 13, . . .Look at the pattern that the common difference of 2 creates.
General Way to Write an Arithmetic SequenceArithmeticSequencesMath 34: Fall2016Consider the Arithmetic Sequence below. Notice the firstterm is 5 and the common difference is 2:ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework5, 7, 9, 11, 13, . . .Look at the pattern that the common difference of 2 creates.5,7,9,11,13,.
General Way to Write an Arithmetic SequenceArithmeticSequencesConsider the Arithmetic Sequence below. Notice the firstterm is 5 and the common difference is 2:Math 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework5, 7, 9, 11, 13, . . .Look at the pattern that the common difference of 2 creates.5,7,9,11,13,.5 , 5 (1)2, 5 (2)2, 5 (3)2, 5 (4)2, . . . {z} {z } {z } {z } {z }a1a2a3a4a5
General Way to Write an Arithmetic SequenceArithmeticSequencesConsider the Arithmetic Sequence below. Notice the firstterm is 5 and the common difference is 2:Math 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework5, 7, 9, 11, 13, . . .Look at the pattern that the common difference of 2 creates.5,7,9,11,13,.5 , 5 (1)2, 5 (2)2, 5 (3)2, 5 (4)2, . . . {z} {z } {z } {z } {z }a1a2a3a4a5We notice the pattern for this sequence an 5 (n 1)2
General Way to Write an Arithmetic SequenceArithmeticSequencesConsider the Arithmetic Sequence below. Notice the firstterm is 5 and the common difference is 2:Math 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamples5, 7, 9, 11, 13, . . .Look at the pattern that the common difference of 2 creates.5,7,9,11,13,.5 , 5 (1)2, 5 (2)2, 5 (3)2, 5 (4)2, . . . {z} {z } {z } {z } {z }a1a2a3a4a5We notice the pattern for this sequence an 5 (n 1)2HomeworkWe also see that an an 1 2(each term is 2 more than the previous term)
General Way to Write an Arithmetic SequenceArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsWay to Write a Formula for an Arithmetic Sequence:Given that a1 , a2 , a3 , . . . is an arithmetic sequence withcommon difference d,We can rewrite the sequence asan a1 (n 1)dwhere the index starts at n 1.FormulaExamplesHomeworkHere a1 is the first term of the sequence (a constant) andd is the common difference (also a constant).
Examples (Arithmetic Sequences)ArithmeticSequencesGiven the Arithmetic Sequence 10, 4, 2, 8, . . .Math 34: Fall2016ArithmeticSequencesReviewReal WorldExamples1Find the fifth term in the sequence.2Find the 20th term in the sequence.3Find a formula for the nth term in the sequence.General Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework
Examples (Arithmetic Sequences)ArithmeticSequencesMath 34: Fall2016Given the Arithmetic Sequence 10, 4, 2, 8, . . .To understand everything about this sequences we need to know:ArithmeticSequencesReviewReal WorldExamples1Find the fifth term in the sequence.2Find the 20th term in the sequence.3Find a formula for the nth term in the sequence.General Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework
Examples (Arithmetic Sequences)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGiven the Arithmetic Sequence 10, 4, 2, 8, . . .To understand everything about this sequences we need to know:It’s ArithmeticWith common difference d 6And first term a1 101Find the fifth term in the sequence.2Find the 20th term in the sequence.3Find a formula for the nth term in the sequence.General Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework
Examples (Arithmetic Sequences)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGiven the Arithmetic Sequence 10, 4, 2, 8, . . .To understand everything about this sequences we need to know:It’s ArithmeticWith common difference d 6And first term a1 101General Wayto Write anArithmeticSequenced 6, we can see the 5th term 6 more than 4th term.FormulaExamplesPartial SumsFind the fifth term in the sequence.Since the first 4 terms are given, and the common difference is2Find the 20th term in the sequence.3Find a formula for the nth term in the sequence.FormulaExamplesHomework
Examples (Arithmetic Sequences)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGiven the Arithmetic Sequence 10, 4, 2, 8, . . .To understand everything about this sequences we need to know:It’s ArithmeticWith common difference d 6And first term a1 101General Wayto Write anArithmeticSequenced 6, we can see the 5th term 6 more than 4th term.i.e. a5 a4 6 8 6 14FormulaExamplesPartial SumsFind the fifth term in the sequence.Since the first 4 terms are given, and the common difference is2Find the 20th term in the sequence.3Find a formula for the nth term in the sequence.FormulaExamplesHomework
Examples (Arithmetic Sequences)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGiven the Arithmetic Sequence 10, 4, 2, 8, . . .To understand everything about this sequences we need to know:It’s ArithmeticWith common difference d 6And first term a1 101General Wayto Write anArithmeticSequenced 6, we can see the 5th term 6 more than 4th term.i.e. a5 a4 6 8 6 14FormulaExamplesPartial SumsFind the fifth term in the sequence.Since the first 4 terms are given, and the common difference is2FormulaExamplesFind the 20th term in the sequence.Use the formula: an a1 (n 1)dHomework3Find a formula for the nth term in the sequence.
Examples (Arithmetic Sequences)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGiven the Arithmetic Sequence 10, 4, 2, 8, . . .To understand everything about this sequences we need to know:It’s ArithmeticWith common difference d 6And first term a1 101General Wayto Write anArithmeticSequenced 6, we can see the 5th term 6 more than 4th term.i.e. a5 a4 6 8 6 14FormulaExamplesPartial SumsFind the fifth term in the sequence.Since the first 4 terms are given, and the common difference is2FormulaExamplesFind the 20th term in the sequence.Use the formula: an a1 (n 1)dan 10 (n 1)6 with starting term n 1Homework3Find a formula for the nth term in the sequence.
Examples (Arithmetic Sequences)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGiven the Arithmetic Sequence 10, 4, 2, 8, . . .To understand everything about this sequences we need to know:It’s ArithmeticWith common difference d 6And first term a1 101General Wayto Write anArithmeticSequenced 6, we can see the 5th term 6 more than 4th term.i.e. a5 a4 6 8 6 14FormulaExamplesPartial SumsFind the fifth term in the sequence.Since the first 4 terms are given, and the common difference is2FormulaExamplesFind the 20th term in the sequence.Use the formula: an a1 (n 1)dan 10 (n 1)6 with starting term n 1HomeworkThis mean the 20th term is: a20 10 (20 1)6 1043Find a formula for the nth term in the sequence.
Examples (Arithmetic Sequences)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGiven the Arithmetic Sequence 10, 4, 2, 8, . . .To understand everything about this sequences we need to know:It’s ArithmeticWith common difference d 6And first term a1 101General Wayto Write anArithmeticSequenced 6, we can see the 5th term 6 more than 4th term.i.e. a5 a4 6 8 6 14FormulaExamplesPartial SumsFind the fifth term in the sequence.Since the first 4 terms are given, and the common difference is2FormulaExamplesFind the 20th term in the sequence.Use the formula: an a1 (n 1)dan 10 (n 1)6 with starting term n 1HomeworkThis mean the 20th term is: a20 10 (20 1)6 1043Find a formula for the nth term in the sequence.Done above because shortcuts are awesome
Examples, Real World Arithmetic Sequences(Number 1)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkJoan invests 3, 000 in an account that pays 2% simpleinterest. Determine how much money is in her account aftereach of the first 5 years.
Examples, Real World Arithmetic Sequences(Number 1)ArithmeticSequencesMath 34: Fall2016Joan invests 3, 000 in an account that pays 2% simpleinterest. Determine how much money is in her account aftereach of the first 5 years.ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkUsing I PRT formula for simple interest.
Examples, Real World Arithmetic Sequences(Number 1)ArithmeticSequencesMath 34: Fall2016Joan invests 3, 000 in an account that pays 2% simpleinterest. Determine how much money is in her account aftereach of the first 5 years.ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceUsing I PRT formula for simple interest.P 3, 000R 0.02T (depends which year we’re talking about)FormulaExamplesPartial SumsFormulaExamplesHomeworkYear1Interest (I PRT) 3000 · 0.02 · 1 60Total In Account 3000 60 3060
Examples, Real World Arithmetic Sequences(Number 1)ArithmeticSequencesMath 34: Fall2016Joan invests 3, 000 in an account that pays 2% simpleinterest. Determine how much money is in her account aftereach of the first 5 years.ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceUsing I PRT formula for simple interest.P 3, 000R 0.02T (depends which year we’re talking about)FormulaExamplesPartial SumsFormulaExamplesHomeworkYear12Interest (I PRT) 3000 · 0.02 · 1 60 3000 · 0.02 · 2 120Total In Account 3000 60 3060 3000 120 3120
Examples, Real World Arithmetic Sequences(Number 1)ArithmeticSequencesMath 34: Fall2016Joan invests 3, 000 in an account that pays 2% simpleinterest. Determine how much money is in her account aftereach of the first 5 years.ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceUsing I PRT formula for simple interest.P 3, 000R 0.02T (depends which year we’re talking about)FormulaExamplesPartial SumsFormulaExamplesHomeworkYear12345Interest (I PRT) 3000 · 0.02 · 1 60 3000 · 0.02 · 2 120 3000 · 0.02 · 3 180 3000 · 0.02 · 4 240 3000 · 0.02 · 5 300Total In Account 3000 60 3060 3000 120 3120 3000 180 3180 3000 240 3240 3000 300 3300
Examples, Real World Arithmetic Sequences(Number 1) Cont.ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkAnother way to think about it:
Examples, Real World Arithmetic Sequences(Number 1) Cont.ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkAnother way to think about it:The simple interest from each year is 3000 · 0.02 · 1 60,so each year Joan has 60 more than the previous year.
Examples, Real World Arithmetic Sequences(Number 1) Cont.ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesAnother way to think about it:The simple interest from each year is 3000 · 0.02 · 1 60,so each year Joan has 60 more than the previous year.ReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkThis looks like an arithmetic sequence.With starting value a1 3060 and common difference d 60.
Examples, Real World Arithmetic Sequences(Number 1) Cont.ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesAnother way to think about it:The simple interest from each year is 3000 · 0.02 · 1 60,so each year Joan has 60 more than the previous year.ReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkThis looks like an arithmetic sequence.With starting value a1 3060 and common difference d 60. 3060, 3120, 3180, 3240, 3300, . . .
Examples, Real World Arithmetic Sequences(Number 1) Cont.ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesAnother way to think about it:The simple interest from each year is 3000 · 0.02 · 1 60,so each year Joan has 60 more than the previous year.ReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsThis looks like an arithmetic sequence.With starting value a1 3060 and common difference d 60. 3060, 3120, 3180, 3240, 3300, . . .FormulaExamplesHomeworkSo the amount of money in the account at (the end of)an 3060 (n 1) 60year n is:
Examples, Real World Arithmetic Sequences(Number 2)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkYou start a new job and you’re told you salary is 29, 000 forthe first year, and that you’ll get a 1700 raise each year. Whatwill your salary be in the third year? What will your salary be in10 years? How long does it take for your salary to (at least)double?
Examples, Real World Arithmetic Sequences(Number 2)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkYou start a new job and you’re told you salary is 29, 000 forthe first year, and that you’ll get a 1700 raise each year. Whatwill your salary be in the third year? What will your salary be in10 years? How long does it take for your salary to (at least)double?Fill in the table indicating your salary in the first severalyears:Year Salaryin indicated year1234
Examples, Real World Arithmetic Sequences(Number 2)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkYou start a new job and you’re told you salary is 29, 000 forthe first year, and that you’ll get a 1700 raise each year. Whatwill your salary be in the third year? What will your salary be in10 years?Fill in the table indicating your salary in the first severalyears:Year Salaryin indicated year1 29, 000234
Examples, Real World Arithmetic Sequences(Number 2)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkYou start a new job and you’re told you salary is 29, 000 forthe first year, and that you’ll get a 1700 raise each year. Whatwill your salary be in the third year? What will your salary be in10 years? How long does it take for your salary to (at least)double?Fill in the table indicating your salary in the first severalyears:Year Salaryin indicated year1 29, 0002 30, 7003 32, 4004 34, 100
Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesNotice that the list of your salaries year by year look likean Arithmetic Sequence.Identify the common difference, and the first term:ReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceWrite a formula for an (your salary in year n).FormulaExamplesPartial SumsFormulaExamplesWhat will your salary be in the third year?HomeworkWhat will your salary be in 10 years?
Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceNotice that the list of your salaries year by year look likean Arithmetic Sequence.Identify the common difference, and the first term:a1 29, 000d 1700(This is the important bit. You make the table to help you with this.)Write a formula for an (your salary in year n).FormulaExamplesPartial SumsFormulaExamplesWhat will your salary be in the third year?HomeworkWhat will your salary be in 10 years?
Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesNotice that the list of your salaries year by year look likean Arithmetic Sequence.Identify the common difference, and the first term:a1 29, 000d 1700(This is the important bit. You make the table to help you with this.)Write a formula for an (your salary in year n).an 29, 000 (n 1) 1700Where n is measured in years, and an is your salary in year n(measured in dollars)What will your salary be in the third year?HomeworkWhat will your salary be in 10 years?
Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkNotice that the list of your salaries year by year look likean Arithmetic Sequence.Identify the common difference, and the first term:a1 29, 000d 1700(This is the important bit. You make the table to help you with this.)Write a formula for an (your salary in year n).an 29, 000 (n 1) 1700Where n is measured in years, and an is your salary in year n(measured in dollars)What will your salary be in the third year?a3 29, 000 (3 1) 1700 32, 400What will your salary be in 10 years?
Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkNotice that the list of your salaries year by year look likean Arithmetic Sequence.Identify the common difference, and the first term:a1 29, 000d 1700(This is the important bit. You make the table to help you with this.)Write a formula for an (your salary in year n).an 29, 000 (n 1) 1700Where n is measured in years, and an is your salary in year n(measured in dollars)What will your salary be in the third year?a3 29, 000 (3 1) 1700 32, 400What will your salary be in 10 years?a10 29, 000 (10 1) 1700 44, 300
Partial Sums of an Arithmetic SequenceArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkRemember Sn is the sum of the first n terms of a sequence.
Partial Sums of an Arithmetic SequenceArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkRemember Sn is the sum of the first n terms of a sequence.Let’s work our a formula for the nth partial sum of anArithmetic Sequence
Partial Sums of an Arithmetic SequenceArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkHere’s one way to write our Arithmetic sequence:a1 , (a1 d), (a1 2d), (a1 3d), . . .
Partial Sums of an Arithmetic SequenceArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkHere’s one way to write our Arithmetic sequence:a1 , (a1 d), (a1 2d), (a1 3d), . . .So the nth Partial Sum of the Arithmetic Series can bewritten as:Sn a1 (a1 d) (a1 2d) · · · (a1 (n 2)d) (a1 (n 1)d){z} an
Partial Sums of an Arithmetic SequenceArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkHere’s one way to write our Arithmetic sequence:a1 , (a1 d), (a1 2d), (a1 3d), . . .So the nth Partial Sum of the Arithmetic Series can bewritten as:Sn a1 (a1 d) (a1 2d) · · · (a1 (n 2)d) (a1 (n 1)d){z} anAnother way to name the terms:a1 {z}an (n 1)d (a1 d) (a1 2d) · · · (a1 (n 2)d) (a1 (n 1)d) {z } {z } {z} {z}an (n 2)dan (n 3)dan dan
Partial Sums of an Arithmetic SequenceArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkHere’s one way to write our Arithmetic sequence:a1 , (a1 d), (a1 2d), (a1 3d), . . .So the nth Partial Sum of the Arithmetic Series can bewritten as:Sn a1 (a1 d) (a1 2d) · · · (a1 (n 2)d) (a1 (n 1)d){z} anAnother way to name the terms:a1 {z}an (n 1)d (a1 d) (a1 2d) · · · (a1 (n 2)d) (a1 (n 1)d) {z } {z } {z} {z}an (n 2)dan (n 3)dan danThis gives us another way to write SnSn an (an d) (an 2d) · · · (an (n 2)d) (an (n 1)d)
Partial Sums of an Arithmetic SequenceArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkHere’s one way to write our Arithmetic sequence:a1 , (a1 d), (a1 2d), (a1 3d), . . .So the nth Partial Sum of the Arithmetic Series can bewritten as:Sn a1 (a1 d) (a1 2d) · · · (a1 (n 2)d) (a1 (n 1)d){z} anAnother way to name the terms:a1 {z}an (n 1)d (a1 d) (a1 2d) · · · (a1 (n 2)d) (a1 (n 1)d) {z } {z } {z} {z}an (n 2)dan (n 3)dan danThis gives us another way to write SnSn an (an d) (an 2d) · · · (an (n 2)d) (an (n 1)d)Add the two ways to write Sn together.
Partial Sums of an Arithmetic Sequence (Cont.)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkAdd the two ways to write Sn together.Sn Sn2Sn a1 (a1 d) (a1 2d)an (an d) (an 2d)(a1 an ) (a1 an ) (a1 an ) · · · (a1 (n 2)d) (a1 (n 1)d) · · · (an (n 2)d) (an (n 1)d) · · · (a1 an ) (a1 an )
Partial Sums of an Arithmetic Sequence (Cont.)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkAdd the two ways to write Sn together.Sn Sn2Sn a1 (a1 d) (a1 2d)an (an d) (an 2d)(a1 an ) (a1 an ) (a1 an ) · · · (a1 (n 2)d) (a1 (n 1)d) · · · (an (n 2)d) (an (n 1)d) · · · (a1 an ) (a1 an )We count the (a1 an ) terms on the right.
Partial Sums of an Arithmetic Sequence (Cont.)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkAdd the two ways to write Sn together.Sn Sn2Sn a1 (a1 d) (a1 2d)an (an d) (an 2d)(a1 an ) (a1 an ) (a1 an ) · · · (a1 (n 2)d) (a1 (n 1)d) · · · (an (n 2)d) (an (n 1)d) · · · (a1 an ) (a1 an )We count the (a1 an ) terms on the right.We see that 2Sn n(a1 an ) and.
Partial Sums of an Arithmetic Sequence (Cont.)ArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesAdd the two ways to write Sn together.Sn Sn2Sn a1 (a1 d) (a1 2d)an (an d) (an 2d)(a1 an ) (a1 an ) (a1 an ) · · · (a1 (n 2)d) (a1 (n 1)d) · · · (an (n 2)d) (an (n 1)d) · · · (a1 an ) (a1 an )We count the (a1 an ) terms on the right.We see that 2Sn n(a1 an ) and.Partial SumsFormulaExamplesHomeworkSn n(a1 an )n (a1 an )22
Formula the nth Partial Sum of an ArithmeticSequenceArithmeticSequencesMath 34: Fall2016ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkThe nth partial sum of an Arithmetic Sequencea1 , a2 , a3 , . . . is given byn(a1 an )2Where a1 is the first term of the Arithmetic Sequence andan is the nth term of the Arithmetic Series.Sn
Using the Formula the nth Partial Sum of anArithmetic SequenceArithmeticSequencesMath 34: Fall2016For the Arithmetic Sequence 7, 10, 13, 16, 19, 22, . . .1Find the 4th Partial Sum of the Sequence.2Find the 20th term of the Sequence3Find the 20th Partial Sum of the SequenceArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework
Using the Formula the nth Partial Sum of anArithmetic SequenceArithmeticSequencesMath 34: Fall2016For the Arithmetic Sequence 7, 10, 13, 16, 19, 22, . . .1Find the 4th Partial Sum of the Sequence.S4 24 (a1 a4 ) 24 (7 16) 2(23) 462Find the 20th term of the Sequence3Find the 20th Partial Sum of the SequenceArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework
Using the Formula the nth Partial Sum of anArithmetic SequenceArithmeticSequencesMath 34: Fall2016For the Arithmetic Sequence 7, 10, 13, 16, 19, 22, . . .1ArithmeticSequences(We can double check that 7 10 13 16 46)ReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFind the 4th Partial Sum of the Sequence.S4 24 (a1 a4 ) 24 (7 16) 2(23) 462Find the 20th term of the Sequence3Find the 20th Partial Sum of the SequenceFormulaExamplesPartial SumsFormulaExamplesHomework
Using the Formula the nth Partial Sum of anArithmetic SequenceArithmeticSequencesMath 34: Fall2016For the Arithmetic Sequence 7, 10, 13, 16, 19, 22, . . .1ArithmeticSequences(We can double check that 7 10 13 16 46)ReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFind the 4th Partial Sum of the Sequence.S4 24 (a1 a4 ) 2
Arithmetic Sequence Formula Examples Partial Sums Formula Examples Homework General Way to Write an Arithmetic Sequence Way to Write a Formula for an Arithmetic Sequence: Given that a 1;a 2;a 3;::: is an arithmetic sequence with common di erence d, We can rewrite the sequence as a n a 1 (n 1)d where the index starts at n 1. Here a
3.2 Arithmetic series: the sum of an arithmetic sequence Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life. 7C 7D Arithmetic Sequence Arithmetic Series 6C 6D Arithmetic sequences Arithmetic series 3.1 3.2 Arithmetic sequence Arithmetic s
Algebra I Unit 10: Arithmetic & Geometric Sequences Math Department TEKS: A.12D 2015 - 2016 Arithmetic Sequences Objectives Students will be able to identify if a sequence is arithmetic. Students will be able to determine the value of a specific term. Students will be able to write arithmetic sequences in explicit form.
arithmetic sequence are called arithmetic means. In the sequence below, 38 and 49 are the arithmetic means between 27 and 60. 5, 16, 27, 38 , 49 , 60 760 Chapter 12 Sequences and Series The n th term of an arithmetic sequence with first term a 1 and common difference d is given by a n a 1 (n 1) d . The n th Term of an Arithmetic Sequence .
Arithmetic & Geometric Sequences Worksheet Level 2: Goals: Identify Arithmetic and Geometric Sequences Find the next term in an arithmetic sequence Find the next term in a geometric sequence Practice #1 Does this pattern represent an arithmetic or geometric sequence? Explain. Find how many dots would be in the next figure?
Arithmetic & Geometric Sequences Worksheet Level 2: Goals: Identify Arithmetic and Geometric Sequences Find the next term in an arithmetic sequence Find the next term in a geometric sequence Practice #1 Does this pattern represent an arithmetic or geometric sequence? Explain. Find how many dots would be in the next figure?
arithmetic sequence finds the nth term of an arithmetic sequence lists down the first few terms of an arithmetic sequence given the general term and vice-versa solves word problems involving arithmetic mean applies the concepts of mean and the nth term of an arithmetic sequence
Arithmetic Series: A series whose terms form an arithmetic sequence. 11 4 Arithmetic Series 7 April 27, 2009 Apr 21 4:18 PM. 11 4 Arithmetic Series 8 April 27, 2009 Apr 21 4:19 PM. 11 4 Arithmetic Series . Homework: page 622 (2 32 even, 37, 40) Title: 11-4 Arithmetic Series Subject: SMART Board Interactive Whiteboard Notes
A First Course in Complex Analysis was written for a one-semester undergradu-ate course developed at Binghamton University (SUNY) and San Francisco State University, and has been adopted at several other institutions. For many of our students, Complex Analysis is their first rigorous analysis (if not mathematics) class they take, and this book reflects this very much. We tried to rely on as .
