2. SequencesSequences & Series& Series
SequenceSequence
An ordered list ofAn ordered list of
numbersnumbers
A progression ofA progression of
numbersnumbers
Can be arithmetic,Can be arithmetic,
geometric or neithergeometric or neither
Can be finite orCan be finite or
infiniteinfinite
SeriesSeries
A value you get whenA value you get when
you add up the termsyou add up the terms
of a sequencesof a sequences
Sum of numbers in aSum of numbers in a
sequencesequence
Uses summationUses summation
notationnotation ΣΣ (sigma)(sigma)
3. Try this…Try this…
Find the 10Find the 10thth
term of this sequenceterm of this sequence
2, 5, 8,…2, 5, 8,…
Start by determining the patternStart by determining the pattern
Adding 3 to the previous numberAdding 3 to the previous number
Known as a common differenceKnown as a common difference
Continue the pattern to get the 10Continue the pattern to get the 10thth
termterm
2, 5, 8, 11, 14, 17, 20, 23, 26,…2, 5, 8, 11, 14, 17, 20, 23, 26,…
So, the 10So, the 10thth
term is 29term is 29
4. Try This…Try This…
Write the first 7 terms ofWrite the first 7 terms of aann = 4= 4nn + 9+ 9
aa11 = 13= 13
aa22 = 17= 17
aa33 = 21= 21
aa44 = 25= 25
aa55 = 29= 29
aa66 = 33= 33
aa77 = 37= 37
5. Determining Rules forDetermining Rules for
a Sequencea Sequence
Example:Example:
Determine a rule for theDetermine a rule for the nnth term of theth term of the
sequence: 1, 16, 81, 256,. . .sequence: 1, 16, 81, 256,. . .
When determining a rule for a sequence you need to compare theWhen determining a rule for a sequence you need to compare the
term number to the actual term.term number to the actual term.
For this sequence 81 is the 3For this sequence 81 is the 3rdrd
term, so you need to determine how to get 81term, so you need to determine how to get 81
from 3.from 3.
Rule: aRule: ann = 4^n= 4^n
6. Determining if aDetermining if a
sequence is Geometric,sequence is Geometric,
Arithmetic or NeitherArithmetic or Neither
Watch the following video on ArithmeticWatch the following video on Arithmetic
and Geometric Sequencesand Geometric Sequences
http://www.virtualnerd.com/algebra-2/sequenhttp://www.virtualnerd.com/algebra-2/sequen
7. Determine if theDetermine if the
following sequences arefollowing sequences are
Arithmetic, Geometric, orArithmetic, Geometric, or
NeitherNeitherProblems:Problems:
1.1. 3, 8, 13, 18, 23,…3, 8, 13, 18, 23,…
2.2. 1, 2, 4, 8, 16,…1, 2, 4, 8, 16,…
3.3. 24, 12, 6, 3, 3/2,24, 12, 6, 3, 3/2,
3/4,…3/4,…
4.4. 55, 51, 47, 43, 39,55, 51, 47, 43, 39,
35,…35,…
5.5. 2, 5, 10, 17,…2, 5, 10, 17,…
6.6. 1, 4, 9, 16, 25, 36,1, 4, 9, 16, 25, 36,
……
Answers:Answers:
1.1. Arithmetic, the commonArithmetic, the common
difference is 5.difference is 5.
2.2. Geometric, theGeometric, the
common ratio is 2.common ratio is 2.
3.3. Geometric, theGeometric, the
common ratio is ½.common ratio is ½.
4.4. Arithmetic, the commonArithmetic, the common
difference is -4.difference is -4.
5.5. Neither, no commonNeither, no common
ratio or difference.ratio or difference.
6.6. Neither, no commonNeither, no common
ratio or difference.ratio or difference.
8. Infinite Vs. FiniteInfinite Vs. Finite
InfiniteInfinite
A sequence that goesA sequence that goes
on foreveron forever
Example:Example:
14, 28, 42, 56, 70,…14, 28, 42, 56, 70,…
FiniteFinite
A sequence that hasA sequence that has
an endan end
Example:Example:
1, 3, 9, 27, and 81.1, 3, 9, 27, and 81.
10. Arithmetic RuleArithmetic Rule
aann = a= a11 + (n - 1)d+ (n - 1)d
aa11 is the first term in the sequenceis the first term in the sequence
n is the number of the term you aren is the number of the term you are
trying to determinetrying to determine
d is the common differenced is the common difference
aann is the value of the term that areis the value of the term that are
looking forlooking for
11. Try this….Try this….
Use the arithmetic formula to determine the 100Use the arithmetic formula to determine the 100thth
term ofterm of
the following sequence:the following sequence:
75, 25, -25, -75, -125,…75, 25, -25, -75, -125,…
aa11 = 75= 75
n = 100n = 100
d = -50d = -50
aann == aa11 + (+ (nn - 1)- 1)dd
== 7575 + (+ (100100 – 1)(– 1)(-50-50))
= -4875= -4875
12. Example of a ArithmeticExample of a Arithmetic
Sequence in the RealSequence in the Real
WorldWorld
Suppose you are training to run a 6 mileSuppose you are training to run a 6 mile
race. You plan to start your training byrace. You plan to start your training by
running 2 miles a week, and then yourunning 2 miles a week, and then you
plan to add a ½ mile more every week.plan to add a ½ mile more every week.
At what week will you be running 6At what week will you be running 6
miles?miles?
13. SolutionSolution
The first term of the sequence will be the initial numberThe first term of the sequence will be the initial number
of miles you plan on running.of miles you plan on running.
The common difference of the sequence will be the ½The common difference of the sequence will be the ½
mile that you increase every week.mile that you increase every week.
n will stand for the number of weeks it will take you ton will stand for the number of weeks it will take you to
reach 6 miles.reach 6 miles.
aann == aa11 + (n - 1)+ (n - 1)dd
66 == 22 + (n – 1)(+ (n – 1)(1/21/2))
14. Geometric RuleGeometric Rule
aann = a= a11*r*r(n-1)(n-1)
aa11 is the 1is the 1stst
term of the sequenceterm of the sequence
aann is the value of the term that areis the value of the term that are
looking forlooking for
n is the number of the term you aren is the number of the term you are
trying to determinetrying to determine
r is the common ratio between termsr is the common ratio between terms
15. Try this…Try this…
Use the geometric rule to determine theUse the geometric rule to determine the
1010thth
term of this sequence:term of this sequence:
4, 20, 100, 5004, 20, 100, 500
aa11 = 4= 4
n = 10n = 10
r = 20/4 = 5r = 20/4 = 5
aann == aa11**rr((nn-1)-1)
== 44 ** 55((1010-1)-1)
=7812500=7812500
16. Example of a GeometricExample of a Geometric
Sequence in the RealSequence in the Real
WorldWorld
Suppose you borrow $10,000Suppose you borrow $10,000
from a bank that charges 5%from a bank that charges 5%
interest. You want to determineinterest. You want to determine
how much you will owe the bankhow much you will owe the bank
over a period of 5 years.over a period of 5 years.
17. SolutionSolution
The first term in the sequences will beThe first term in the sequences will be
the initial amount of money borrowed,the initial amount of money borrowed,
which is $10,000.which is $10,000.
The common ratio is 105%, this can beThe common ratio is 105%, this can be
represented as 1.05 as a decimal.represented as 1.05 as a decimal.
n is the number of years you have then is the number of years you have the
loan.loan.
aann = $10,000(1.05)= $10,000(1.05)(5-1)(5-1)
19. Works CitedWorks Cited
Geometric Sequences in the Real World.Geometric Sequences in the Real World. SophiaSophia. N.p., n.d. Web. 11 May 2013.. N.p., n.d. Web. 11 May 2013.
McDougal Littell ClassZone.McDougal Littell ClassZone. ClassZoneClassZone. N.p., n.d. Web. 11 May 2013.. N.p., n.d. Web. 11 May 2013.
There Is Video.Then, There Is Virtual Nerd.There Is Video.Then, There Is Virtual Nerd. How Do You Determine If a Sequence IsHow Do You Determine If a Sequence Is
Arithmetic or Geometric?Arithmetic or Geometric? N.p., n.d. Web. 11 May 2013.N.p., n.d. Web. 11 May 2013.
