12.1 an introduction to sequences & series. sequence: a list of ordered numbers separated by...
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12.1 An Introduction to 12.1 An Introduction to Sequences & SeriesSequences & Series
SequenceSequence::• A list of ordered numbers separated by A list of ordered numbers separated by
commas. commas. • Each number in the list is called a Each number in the list is called a termterm..• For Example:For Example:
Sequence 1Sequence 1 Sequence 2Sequence 2 2,4,6,8,102,4,6,8,10 2,4,6,8,10,… 2,4,6,8,10,…
Term 1, 2, 3, 4, 5Term 1, 2, 3, 4, 5 Term 1, 2, 3, 4, 5Term 1, 2, 3, 4, 5DomainDomain – relative position of each term (1,2,3,4,5) – relative position of each term (1,2,3,4,5)
Usually begins with position 1 unless otherwise Usually begins with position 1 unless otherwise stated.stated.
RangeRange – the actual terms of the sequence – the actual terms of the sequence (2,4,6,8,10)(2,4,6,8,10)
Sequence 1Sequence 1 Sequence 2Sequence 2
2,4,6,8,102,4,6,8,10 2,4,6,8,10,…2,4,6,8,10,…
A sequence can be A sequence can be finitefinite or or infiniteinfinite..
The sequence has The sequence has a last term or a last term or finalfinal
term.term.
(such as seq. 1)(such as seq. 1)
The sequence The sequence continues without continues without
stopping.stopping.
(such as seq. 2)(such as seq. 2)Both sequences have a Both sequences have a general rulegeneral rule: a: ann = 2n where = 2n where
n is the term # and an is the term # and ann is the nth term. is the nth term.
The general rule can also be written in The general rule can also be written in function function notationnotation: f(n) = 2n: f(n) = 2n
Examples:Examples:• Write the first 6 Write the first 6
terms of aterms of ann=5-n.=5-n.
• aa11=5-1=4=5-1=4
• aa22=5-2=3=5-2=3
• aa33=5-3=2=5-3=2
• aa44=5-4=1=5-4=1
• aa55=5-5=0=5-5=0
• aa66=5-6=-1=5-6=-1
• 4,3,2,1,0,-14,3,2,1,0,-1
• Write the first 6 Write the first 6 terms of aterms of ann=2=2nn..
• aa11=2=211=2=2
• aa22=2=222=4=4
• aa33=2=233=8=8
• aa44=2=244=16=16
• aa55=2=255=32=32
• aa66=2=266=64=64
• 2,4,8,16,32,642,4,8,16,32,64
ExamplesExamples: Write a rule for the nth term.: Write a rule for the nth term.
The seq. can be The seq. can be written as:written as:
Or, aOr, ann=2/(5=2/(5nn))
• The seq. can be The seq. can be written as:written as:
2(1)+1, 2(2)+1, 2(3)+1, 2(1)+1, 2(2)+1, 2(3)+1, 2(4)+1,…2(4)+1,…
Or, aOr, ann=2n+1=2n+1
,...625
2,
125
2,
25
2,
5
2 .a
,...5
2,
5
2,
5
2,
5
24321
,...9,7,5,3 .b
Example: write a rule for the nth term.
• 2,6,12,20,…2,6,12,20,…
• Can be written as:Can be written as:
1(2), 2(3), 3(4), 4(5),…1(2), 2(3), 3(4), 4(5),…
Or, aOr, ann=n(n+1)=n(n+1)
Graphing a SequenceGraphing a Sequence• Think of a sequence as ordered pairs for Think of a sequence as ordered pairs for
graphing. (n , agraphing. (n , ann))
• For example: 3,6,9,12,15 For example: 3,6,9,12,15 would be the ordered pairs (1,3), (2,6), would be the ordered pairs (1,3), (2,6), (3,9), (4,12), (5,15) graphed like points in a (3,9), (4,12), (5,15) graphed like points in a scatter plotscatter plot
* Sometimes it helps to find the rule first * Sometimes it helps to find the rule first when you are not given every term in a when you are not given every term in a finite sequence.finite sequence.
Term #Term # Actual termActual term
Graphing
na
1
3
2
6
3
9
4
12
• What is a sequence?A collections of objects that is ordered so that
there is a 1st, 2nd, 3rd,… member.• What is the difference between finite and
infinite?
Finite means there is a last term. Infinite means the sequence continues without stopping.
Sequences and Series• What is a series?• How do you know the difference between a How do you know the difference between a
sequence and a series?sequence and a series?
• What is sigma notation?• How do you write a series with summation
notation?• Name 3 formulas for special series.
SeriesSeries• The sum of the terms in a sequence.The sum of the terms in a sequence.
• Can be finite or infiniteCan be finite or infinite
• For Example:For Example:
Finite Seq.Finite Seq. Infinite Seq.Infinite Seq.
2,4,6,8,102,4,6,8,10 2,4,6,8,10,…2,4,6,8,10,…
Finite SeriesFinite Series Infinite SeriesInfinite Series
2+4+6+8+102+4+6+8+10 2+4+6+8+10+…2+4+6+8+10+…
Summation NotationSummation Notation• Also called Also called sigma notationsigma notation
(sigma is a Greek letter (sigma is a Greek letter ΣΣ meaning “sum”) meaning “sum”)
The series 2+4+6+8+10 can be written as:The series 2+4+6+8+10 can be written as:
i is called the i is called the index of summationindex of summation
(it’s just like the n used earlier). (it’s just like the n used earlier).
Sometimes you will see an n or k here instead of i.Sometimes you will see an n or k here instead of i.
The notation is read:The notation is read:
““the sum from i=1 to 5 of 2i”the sum from i=1 to 5 of 2i”
5
1
2ii goes from 1 i goes from 1
to 5.to 5.
Summation Notation for an Summation Notation for an Infinite SeriesInfinite Series
• Summation notation for the infinite series:Summation notation for the infinite series:
2+4+6+8+10+… would be written as:2+4+6+8+10+… would be written as:
Because the series is infinite, you must use i Because the series is infinite, you must use i from 1 to infinity (from 1 to infinity (∞) instead of stopping at ∞) instead of stopping at
the 5the 5thth term like before. term like before.
1
2i
Examples: Write each series in Examples: Write each series in summation notation.summation notation.
a. 4+8+12+…+100a. 4+8+12+…+100• Notice the series can Notice the series can
be written as:be written as:
4(1)+4(2)+4(3)+…+4(25)4(1)+4(2)+4(3)+…+4(25)
Or 4(i) where i goes Or 4(i) where i goes from 1 to 25.from 1 to 25.
• Notice the series Notice the series can be written as:can be written as:
25
1
4i
...5
4
4
3
3
2
2
1 . b
...14
4
13
3
12
2
11
1
. to1 from goes where1
Or,
ii
i
1 1i
i
ExampleExample: Find the sum of the : Find the sum of the series.series.
• k goes from 5 to 10.k goes from 5 to 10.
• (5(522+1)+(6+1)+(622+1)+(7+1)+(722+1)+(8+1)+(822+1)+(9+1)+(922+1)+(10+1)+(1022+1)+1)
= 26+37+50+65+82+101= 26+37+50+65+82+101
= = 361361
10
5
2 1k
Special Formulas (shortcuts!)Special Formulas (shortcuts!)
nn
i
1
12
)1(
1
nni
n
i
6
)12)(1(
1
2
nnni
n
i
1
n
i
c cn
Example: Find the sum.Example: Find the sum.
• Use the 3Use the 3rdrd shortcut! shortcut!
10
1
2
i
i
6
)12)(1( nnn
6
)110*2)(110(10
6
21*11*10 385
6
2310
• What is a series?A series occurs when the terms of a sequence are
added.• How do you know the difference between a
sequence and a series?The plus signs• What is sigma notation?∑• How do you write a series with summation
notation?Use the sigma notation with the pattern rule.• Name 3 formulas for special series.
1
n
i
c cn
2
)1(
1
nni
n
i 6
)12)(1(
1
2
nnni
n
i
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