Math 34: Arithmetic Sequences

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ArithmeticSequencesMath 34:Spring 2015ArithmeticSequencesArithmetic SequencesSupplemental Material Not Found in You TextReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceMath 34: Spring 2015FormulaExamplesPartial SumsFormulaExamplesDo NOT print these slides!!There are printer friendly files on the website.HomeworkJanuary 28, 2015

Arithmetic SequencesArithmeticSequencesMath 34:Spring 2015ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework1 Arithmetic SequencesReviewReal World Examples2 General Way to Write an Arithmetic SequenceFormulaExamples3 Partial SumsFormulaExamples4 Homework

Arithmetic SequencesArithmeticSequencesMath 34:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesRecall 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?11, 4, 7, 10, 13, . . .22, 4, 8, 16, 32, . . .3 3, 7, 17, 27, . . .Partial SumsFormulaExamplesHomework

Arithmetic SequencesArithmeticSequencesMath 34:Spring 2015ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesRecall 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?1, 4, 7, 10, 13, . . .IS arithmetic, with constant difference d 32 2, 4, 8, 16, 32, . . .1Homework3 3, 7, 17, 27, . . .

Arithmetic SequencesArithmeticSequencesMath 34:Spring 2015ArithmeticSequencesReviewReal 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?1, 4, 7, 10, 13, . . .IS arithmetic, with constant difference d 32 2, 4, 8, 16, 32, . . .is NOT arithmetic3 3, 7, 17, 27, . . .1

Arithmetic SequencesArithmeticSequencesMath 34:Spring 2015ArithmeticSequencesReviewReal 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?1, 4, 7, 10, 13, . . .IS arithmetic, with constant difference d 32 2, 4, 8, 16, 32, . . .is NOT arithmetic3 3, 7, 17, 27, . . .IS arithmetic, with constant difference d 101

Real World ExamplesArithmeticSequences1You 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? How long does it take for yoursalary to (at least) double?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?Math 34:Spring 2015ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework

Real World ExamplesArithmeticSequences1You 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? How long does it take for yoursalary to (at least) double?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?Math 34:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015Consider 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:Spring 2015Consider 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:Spring 2015Consider 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:Spring 2015ArithmeticSequences5, 7, 9, 11, 13, . . .ReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkLook 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:Spring 2015ArithmeticSequences5, 7, 9, 11, 13, . . .ReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkLook 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:Spring 2015ArithmeticSequences5, 7, 9, 11, 13, . . .ReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesLook 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 theprevious term)

General Way to Write an Arithmetic SequenceArithmeticSequencesMath 34:Spring 2015ArithmeticSequencesWay to Write a Formula for an Arithmetic Sequence:Given that a1 , a2 , a3 , . . . is an arithmetic sequence withcommon difference d,ReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsWe 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015Given 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015Joan 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:Spring 2015Joan 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:Spring 2015Joan 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:Spring 2015Joan 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:Spring 2015ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkAnother way to think about it:

Examples, Real World Arithmetic Sequences(Number 1) Cont.ArithmeticSequencesMath 34:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesAnother 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:Spring 2015ArithmeticSequencesAnother 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:Spring 2015ArithmeticSequencesAnother 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 3000 (n 1) 60year n is:

Examples, Real World Arithmetic Sequences(Number 2)ArithmeticSequencesMath 34:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal 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, 000234

Examples, Real World Arithmetic Sequences(Number 2)ArithmeticSequencesMath 34:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesNotice 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal 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:Spring 2015ArithmeticSequencesReviewReal 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

Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34:Spring 2015ArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomeworkHow long does it take for your salary to (at least) double?

Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34:Spring 2015How long does it take for your salary to (at least) double?Double your (starting) salary is 58, 000This is a value for anArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework

Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34:Spring 2015How long does it take for your salary to (at least) double?Double your (starting) salary is 58, 000This is a value for anArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework 58, 000 29, 000 (n 1) 1700

Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34:Spring 2015How long does it take for your salary to (at least) double?Double your (starting) salary is 58, 000This is a value for anArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework 58, 000 29, 000 (n 1) 1700 58, 000 29, 000 (n 1) 1700

Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34:Spring 2015How long does it take for your salary to (at least) double?Double your (starting) salary is 58, 000This is a value for anArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework 58, 000 29, 000 (n 1) 1700 58, 000 29, 000 (n 1) 1700 29, 000 n · 1700 1700

Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34:Spring 2015How long does it take for your salary to (at least) double?Double your (starting) salary is 58, 000This is a value for anArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework 58, 000 58, 000 29, 000 29, 000 29, 000 1700 29, 000 (n 1) 1700(n 1) 1700n · 1700 1700n · 1700

Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34:Spring 2015How long does it take for your salary to (at least) double?Double your (starting) salary is 58, 000This is a value for anArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework 58, 000 58, 000 29, 000 29, 000 29, 000 1700 30, 700 29, 000 (n 1) 1700(n 1) 1700n · 1700 1700n · 1700n · 1700

Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34:Spring 2015How long does it take for your salary to (at least) double?Double your (starting) salary is 58, 000This is a value for anArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework 58, 000 58, 000 29, 000 29, 000 29, 000 1700 30, 700 30,700 1700 29, 000 (n 1) 1700(n 1) 1700n · 1700 1700n · 1700n · 1700n

Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34:Spring 2015How long does it take for your salary to (at least) double?Double your (starting) salary is 58, 000This is a value for anArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework 58, 000 58, 000 29, 000 29, 000 29, 000 1700 30, 700 30,700 1700 29, 000 (n 1) 1700(n 1) 1700n · 1700 1700n · 1700n · 1700nSo n 18.0588, We must round up to 19 years.

Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34:Spring 2015How long does it take for your salary to (at least) double?Double your (starting) salary is 58, 000This is a value for anArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework 58, 000 58, 000 29, 000 29, 000 29, 000 1700 30, 700 30,700 1700 29, 000 (n 1) 1700(n 1) 1700n · 1700 1700n · 1700n · 1700nSo n 18.0588, We must round up to 19 years.Double Check:

Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34:Spring 2015How long does it take for your salary to (at least) double?Double your (starting) salary is 58, 000This is a value for anArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework 58, 000 58, 000 29, 000 29, 000 29, 000 1700 30, 700 30,700 1700 29, 000 (n 1) 1700(n 1) 1700n · 1700 1700n · 1700n · 1700nSo n 18.0588, We must round up to 19 years.Double Check:a18 29, 000 (18 1) 1700 57, 900 (less than double)a19 29, 000 (19 1) 1700 59, 600 (more than double)

Examples, Real World Arithmetic Sequences(Number 2) Cont.ArithmeticSequencesMath 34:Spring 2015How long does it take for your salary to (at least) double?Double your (starting) salary is 58, 000This is a value for anArithmeticSequencesReviewReal WorldExamplesGeneral Wayto Write anArithmeticSequenceFormulaExamplesPartial SumsFormulaExamplesHomework 58, 000 58, 000 29, 000 29, 000 29, 000 1700 30, 700 30,700 1700 29, 000 (n 1) 1700(n 1) 1700n · 1700 1700n · 1700n · 1700nSo n 18.0588, We must round up to 19 years.Double Check:a18 29, 000 (18 1) 1700 57, 900 (less than double)a19

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

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