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Understanding 8.1…. Use sigma notation to write the sum of. Arithmetic Sequences 8.2. JMerrill, 2007 Revised 2008. Sequences. A Sequence: Usually defined to be a function Domain is the set of positive integers Arithmetic sequence graphs are linear (usually). Sequences. - PowerPoint PPT Presentation
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Understanding 8.1…Understanding 8.1…
Use sigma notation to write the sum of Use sigma notation to write the sum of
1 2 6 24 120 720
2 4 8 16 32 64
6
1
! 73
2 4kk
k
Arithmetic SequencesArithmetic Sequences8.28.2
JMerrill, 2007JMerrill, 2007
Revised 2008Revised 2008
SequencesSequences
A Sequence:A Sequence: Usually defined to be a functionUsually defined to be a function Domain is the set of positive integersDomain is the set of positive integers Arithmetic sequence graphs are linear Arithmetic sequence graphs are linear
(usually)(usually)
SequencesSequences
SEQUENCE SEQUENCE - a set of numbers, called - a set of numbers, called terms, arranged in a particular order. terms, arranged in a particular order.
There are two basic types: There are two basic types: ArithmeticArithmetic GeometricGeometric
This unit deals with arithmetic sequencesThis unit deals with arithmetic sequences
Arithmetic SequencesArithmetic Sequences
ARITHMETICARITHMETIC - the difference of any two consecutive - the difference of any two consecutive terms is constant. terms is constant. In order to find the difference, you MUST pick one term In order to find the difference, you MUST pick one term and subtract the and subtract the precedingpreceding term termYouYou MUST MUST check more than 1 pair of terms!check more than 1 pair of terms!
2,6,10,14,18………2,6,10,14,18………difference = 4difference = 4
17,10,3,-4,-11,-18…….17,10,3,-4,-11,-18…….difference = -7difference = -7
a, a+d, a+2d, a+3d………….a, a+d, a+2d, a+3d………….difference = ddifference = d
Are you Ready???Are you Ready???
The difference of 8, 3, -2, -7… The difference of 8, 3, -2, -7… is 5is 5
True or False?True or False?
The difference of 23, 17, 11, 5, is -6The difference of 23, 17, 11, 5, is -6
True or False?True or False?
Formulas for the nFormulas for the nthth term of a term of a SequenceSequence
Arithmetic:Arithmetic: aann = = aa11 + + ((nn-1)-1)dd
To get the To get the nnthth term term, start with the , start with the 11stst term termand add the and add the differencedifference (n-1)(n-1) times times
n = THE TERM NUMBER
ExampleExample
Find a formula for aFind a formula for ann and sketch the graph and sketch the graph for the sequence 1, 4, 7, 10...for the sequence 1, 4, 7, 10...
Arithmetic or Geometric? Arithmetic or Geometric? d = ? d = ?
aann = a = a11 + ( + (nn-1)-1)dd
aann = = 11 + ( + (nn-1)-1)33
aann = 1 + 3 = 1 + 3nn-3-3
aann = -2 + 3n = -2 + 3n
3
n = THE TERM NUMBER
ExampleExample
Find the given term of the arithmetic Find the given term of the arithmetic sequence if asequence if a11 = 15, a = 15, a22 =21, find a =21, find a2020
d = d =
aann = = aa11 + ( + (nn-1)-1)dd
aa2020 = = 1515 + (19 + (19)6)6
aa2020 = 15 + 114 = 15 + 114
aa2020 = 129 = 129
6
You DoYou Do
Find a formula for the nth term of Find a formula for the nth term of an arithmetic sequence whose an arithmetic sequence whose common difference is 3 and common difference is 3 and whose first term is 2whose first term is 2
aann = 3n - 1 = 3n - 1
Last ExampleLast Example
The 4The 4thth term of an arithmetic sequence is 20, term of an arithmetic sequence is 20, and the 13and the 13thth term is 65. Write the first 3 terms of term is 65. Write the first 3 terms of the sequence?the sequence?
11st st useuse the equation:the equation:
aann = = aa11 + ( + (nn-1)-1)dd
aa44 = = aa11 + ( + (44-1)-1)dd aa1313 = = aa11 + ( + (1313-1)-1)dd
20 20 = = aa11 + + 3d3d 65 65 = = aa11 + 12 + 12dd
20 – 3d = a20 – 3d = a11 65 – 12d = a65 – 12d = a11
20 – 3d = 65 – 12d; d = 520 – 3d = 65 – 12d; d = 5
Last ExampleLast Example
Knowing that d = 5 and the 4Knowing that d = 5 and the 4 thth term is 20, we can term is 20, we can subtract 5 each time and know that the subtract 5 each time and know that the sequence is 5, 10, 15, 20…sequence is 5, 10, 15, 20…
If we had been asked to find the equation (and If we had been asked to find the equation (and we couldn’t figure out that the 1we couldn’t figure out that the 1stst term was 5)… term was 5)…
20 20 = = aa11 + + 3d3d
20 20 = = aa11 + + 3(5)3(5)
5 = a5 = a11
So, So, aann = 5 + ( = 5 + (nn-1)5 and -1)5 and aann = 5n = 5n
Understanding ProblemUnderstanding Problem
Write the 1Write the 1stst 5 terms of the sequence. If 5 terms of the sequence. If the sequence is arithmetic, find the the sequence is arithmetic, find the common difference.common difference.
1
1na n
1 1 1 1 1
, , , ,2 3 4 5 6
not arithmetic
Sum of a Finite Arithmetic SeriesSum of a Finite Arithmetic Series
The sum of the 1st n terms of an arithmetic series is
1
1
( )
2
2
nn
n n
n a aS
nThe book uses S a a
You can see that you need the first term and the nth term.
ExampleExample
Find the sum of the 1Find the sum of the 1stst 25 terms of the 25 terms of the arithmetic series 11 + 14 + 17 + 20 + …arithmetic series 11 + 14 + 17 + 20 + …
Step 1: Find the 25Step 1: Find the 25thth term: term:
1
25
( 1)
11 (25 1)3 83na a n d
a
25
25(11 83)1175
2S
ExampleExample
Find the sum of the cubes of the first Find the sum of the cubes of the first twenty positive integers.twenty positive integers.
So, we want SSo, we want S2020 = 1 = 13 3 + 2+ 233 + 3 + 333 + …+ 20 + …+ 2033
aa11 = 1 = 1
aa2020 = 20 = 2033 = 8000 = 8000
20
1
20
( )2
20(1 8000)2
80,010
nn
n a aS
S
S
You DoYou Do
Find the 150Find the 150thth partial sum of the arithmetic partial sum of the arithmetic sequence 5, 16, 27, 38, 49sequence 5, 16, 27, 38, 49
Can you do it?Can you do it?
123,675123,675
Last ExampleLast Example
An auditorium has 20 rows of seats. An auditorium has 20 rows of seats. There are 20 seats in the 1There are 20 seats in the 1stst row, 21 seats row, 21 seats in the 2in the 2ndnd row, 22 seats in the 3 row, 22 seats in the 3rdrd row, and row, and so on. How many seats are there in all 20 so on. How many seats are there in all 20 rows?rows?aa11 = 20 = 20
aa22 = 21 = 21
aa33 = 22 = 22
d = 1d = 1
aann = = aa11 + ( + (nn-1)-1)dd
aa2020 = = 2020 + 19 + 19
aa2020 = 39 = 39 (20 39)590
2n
nS
Last ProblemLast Problem
Find the partial sum of the following Find the partial sum of the following problem WITHOUT a calculator (use problem WITHOUT a calculator (use formula) formula)
250
1
(1000 )n
n
218,625
