11.2 arithmetic sequences & series

11.2 Arithmetic Sequences 11.2 Arithmetic Sequences & Series& Series

p.659p.659

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).

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

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

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

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

• 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.

Assignment

p. 663, 15-43 odd

Arithmetic Sequences and Seriesday 3

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

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

2

1 nn

aanS

Assignment:Assignment:

p. 663p. 663

45-56 all45-56 all

Skip 49 & 50Skip 49 & 50