7.2 Analyze Arithmetic 7.2 Analyze Arithmetic Sequences & SeriesSequences & Series

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What is an arithmetic sequence?What is an arithmetic sequence?

What is the rule for an arithmetic sequence?What is the rule for an arithmetic sequence?

How do you find the rule when given two terms?How do you find the rule when given two terms?

Arithmetic Sequence:Arithmetic Sequence:

• The difference between consecutive The difference between consecutive terms is constant (or the same).terms is constant (or the same).

• The constant difference is also known The constant difference is also known as the as the common difference common difference (d).(d).

Find the common difference by subtracting the term on the left from the next term on the right.

Example: Decide whether each Example: Decide whether each sequence is arithmetic.sequence is arithmetic.

• -10,-6,-2,0,2,6,10,…-10,-6,-2,0,2,6,10,…

• -6--10=4-6--10=4

• -2--6=4-2--6=4

• 0--2=20--2=2

• 2-0=22-0=2

• 6-2=46-2=4

• 10-6=410-6=4

Not arithmetic (because Not arithmetic (because the differences are the differences are not the same)not the same)

• 5,11,17,23,29,…5,11,17,23,29,…

• 11-5=611-5=6

• 17-11=617-11=6

• 23-17=623-17=6

• 29-23=629-23=6

• Arithmetic (commonArithmetic (common difference is 6)difference is 6)

Rule for an Arithmetic SequenceRule for an Arithmetic Sequence

ExampleExample:: Write a rule for the nth Write a rule for the nth term of the sequence 32,47,62,77,… . term of the sequence 32,47,62,77,… .

Then, find a Then, find a1212..

• There is a common difference where d=15, There is a common difference where d=15, therefore the sequence is arithmetic.therefore the sequence is arithmetic.

• Use aUse ann=a=a11+(n-1)d+(n-1)d

aann=32+(n-1)(15) =32+(n-1)(15)

aann=32+15n-15=32+15n-15

aann=17+15n=17+15n

aa1212=17+15(12)=197=17+15(12)=197

One term of an arithmetic sequence is a19 = 48. The common difference is d = 3.

an = a1 + (n – 1) da19 = a1 + (19 – 1) d

48 = a1 + 18(3)

Write general rule.

Substitute 19 for n

Solve for a1.

a.

So, a rule for the nth term is:

a. Write a rule for the nth term.

– 6 = a1

Substitute 48 for a19 and 3 for d.

SOLUTION

a. Use the general rule to find the first term.

an = a1 + (n – 1) d= – 6 + (n – 1) 3

Write general rule.Substitute – 6 for a1 and 3 for d.Simplify.

b. Graph the sequence. One term of an arithmetic sequence is a19 = 48. The common difference is d =3.

Create a table of values for the sequence. The graph of the first 6 terms of the sequence is shown. Notice that the points lie on a line. This is true for any arithmetic sequence.

b.

ExampleExample: One term of an arithmetic sequence : One term of an arithmetic sequence is ais a88=50. The common difference is 0.25. =50. The common difference is 0.25.

Write a rule for the nth term.Write a rule for the nth term.• Use aUse ann=a=a11+(n-1)d to find the 1+(n-1)d to find the 1stst term! term!

aa88=a=a11+(8-1)(.25)+(8-1)(.25)

50=a50=a11+(7)(.25)+(7)(.25)

50=a50=a11+1.75+1.75

48.25=a48.25=a11

* Now, use a* Now, use ann=a=a11+(n-1)d to find the rule.+(n-1)d to find the rule.

aann=48.25+(n-1)(.25)=48.25+(n-1)(.25)

aann=48.25+.25n-.25=48.25+.25n-.25

aann=48+.25n=48+.25n

Now graph an=48+.25n.

• Just like yesterday, remember to graph the Just like yesterday, remember to graph the ordered pairs of the form (n,aordered pairs of the form (n,ann))

• So, graph the points (1,48.25), (2,48.5), So, graph the points (1,48.25), (2,48.5), (3,48.75), (4,49), etc. (3,48.75), (4,49), etc.

Example: Two terms of an arithmetic sequence are Example: Two terms of an arithmetic sequence are aa55=10 and a=10 and a3030=110. Write a rule for the nth term.=110. Write a rule for the nth term.

• Begin by writing 2 equations; one for each term Begin by writing 2 equations; one for each term given.given.

aa55=a=a11+(5-1)d OR 10=a+(5-1)d OR 10=a11+4d+4d

AndAnd

aa3030=a=a11+(30-1)d OR 110=a+(30-1)d OR 110=a11+29d+29d• Now use the 2 equations to solve for aNow use the 2 equations to solve for a11 & d. & d.

10=a10=a11+4d+4d

110=a110=a11+29d (subtract the equations to cancel a+29d (subtract the equations to cancel a11))

-100= -25d -100= -25d

So, d=4 and aSo, d=4 and a11=-6 (now find the rule)=-6 (now find the rule)

aann=a=a11+(n-1)d+(n-1)d

aann=-6+(n-1)(4) OR a=-6+(n-1)(4) OR ann=-10+4n=-10+4n

Example (part 2):Example (part 2): using the rule a using the rule ann=-10+4n, =-10+4n,

write the value of n for which awrite the value of n for which ann=-2.=-2.

-2=-10+4n-2=-10+4n

8=4n8=4n

2=n2=n

Two terms of an arithmetic sequence are a8 = 21 and a27 = 97. Find a rule for the nth term.

SOLUTION

STEP 1Write a system of equations using an = a1 + (n – 1)d and substituting 27 for n (Eq 1) and then 8 for n (Eq 2).

STEP 2 Solve the system. 76 = 19d

4 = d

97 = a1 + 26(4)

Subtract.Solve for d.Substitute for d in Eq 1.

27 = a1 Solve for a1.STEP 3 Find a rule for an.an = a1 + (n – 1)d Write general rule.

= – 7 + (n – 1)4 Substitute for a1 and d.

= – 11 + 4n Simplify.

a27 = a1 + (27 – 1)d 97 = a1 + 26da8 = a1 + (8 – 1)d 21 = a1 + 7d

Equation 1Equation 2

• What is an arithmetic sequence?What is an arithmetic sequence?

The difference between consecutive terms The difference between consecutive terms is a constantis a constant

• What is the rule for an arithmetic What is the rule for an arithmetic sequence?sequence?

aann=a=a11+(n-1)d+(n-1)d

• How do you find the rule when given two How do you find the rule when given two terms?terms?

Write two equations with two unknowns and Write two equations with two unknowns and use linear combination to solve for the use linear combination to solve for the variables.variables.

7.2 Assignment

p. 446, 3-35 odd

Analyze Arithmetic Sequences and Series day 2

What is the formula for find the sum of a finite arithmetic series?

Arithmetic SeriesArithmetic Series• The sum of the The sum of the

terms in an terms in an arithmetic sequencearithmetic sequence

• The formula to find The formula to find the sum of a finite the sum of a finite arithmetic series is:arithmetic series is:

2

1 nn

aanS

# of terms# of terms

11stst Term Term

Last Last TermTerm

ExampleExample: Consider the arithmetic : Consider the arithmetic series 20+18+16+14+… .series 20+18+16+14+… .

• Find the sum of the 1Find the sum of the 1stst 25 terms.25 terms.

• First find the rule for First find the rule for the nth term.the nth term.

• aann=22-2n=22-2n

• So, aSo, a2525 = -28 (last term) = -28 (last term)

• Find n such that SFind n such that Snn=-760=-760

2

1 nn

aanS

2

28202525S 100)4(2525 S

2

1 nn

aanS

2

)222(20760

nn

-1520=n(20+22-2n)-1520=n(20+22-2n)

-1520=-2n-1520=-2n22+42n+42n

2n2n22-42n-1520=0-42n-1520=0

nn22-21n-760=0-21n-760=0

(n-40)(n+19)=0(n-40)(n+19)=0

n=40 or n=-19n=40 or n=-19

Always choose the positive solution!Always choose the positive solution!

2

)222(20760

nn

SOLUTION

a1 = 3 + 5(1) = 8

a20 = 3 + 5(20) =103

S20 = 20( )8 + 103

2= 1110

Identify first term.Identify last term.

Write rule for S20, substituting 8 for a1 and 103 for a20.Simplify.

ANSWER The correct answer is C.

You are making a house of cards similar to the one shown

Write a rule for the number of cards in the nth row if the top row is row 1.

a.

House Of Cards

SOLUTION

Starting with the top row, the numbers of cards in the rows are 3, 6, 9, 12, . . . . These numbers form an arithmetic sequence with a first term of 3 and a common difference of 3. So, a rule for the sequence is:

a.

an = a1 + (n– 1) = d

= 3 + (n – 1)3= 3n

Write general rule.Substitute 3 for a1 and 3 for d.Simplify.

You are making a house of cards similar to the one shown

What is the total number of cards if the house of cards has 14 rows?

b.

House Of Cards

SOLUTION

Total number of cards = S14

Find the sum of an arithmetic series with first term a1 = 3 and last term a14 = 3(14) = 42.

b.

5. Find the sum of the arithmetic series (2 + 7i).i = 1

SOLUTION

a1 = 2 + 7(1) = 9

a12 = 2 + (7)(12) = 2 + 84= 86

( )Sn = n

a1 + an 2

S12 = 570ANSWER S12 = 570

What is the formula for find the sum of a finite arithmetic series?

2

1 nn

aanS

7.2 Assignment:7.2 Assignment:

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40-48 all, 63-6440-48 all, 63-64