aim: summation notation course: alg. 2 & trig. do now: aim: what is this symbol it’s greek to...
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Aim: Summation Notation Course: Alg. 2 & Trig.
Do Now:
Aim: What is this symbol It’s Greek to me!
?
Find the sum of the geometric series.
2 3
3 1, , 5
8 4a a n
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Aim: Summation Notation Course: Alg. 2 & Trig.
Definition of Summation Notation
sigma
sum of terms
The sum of the first n terms of a sequence is represented by
1 2 3 4 n,a a a a a L
When n is a specific number, the sum of a sequence is called a finite series.
1 2 3 41
n
i n,i
a a a a a a
L
where i is the index of summation,n is the upper limit of summation, and1 is the lower limit of summation.
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Aim: Summation Notation Course: Alg. 2 & Trig.
Summation Notation
50 1 2 3 4 5
0
3 3 3 3 3 3 3
= 1 3 9 27 81 243 364
i
i
where i is the index of summation
n is the upper limit of summation
0 is the lower limit of summation
The first term of the summation is formed by substituting the lower limit for the index into the general term. Each succeeding term of the summation is formed using successive integral values of the index, until the upper limit is reached.
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Aim: Summation Notation Course: Alg. 2 & Trig.
Properties of Sums
constantanyis,.111
caccan
ii
n
ii
n
i
n
iii
n
iii baba
1 11
)(.2
n
i
n
iii
n
iii baba
1 11
)(.3
3 3
1 2 31 1
2 2 2i ii i
x x x x x
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Aim: Summation Notation Course: Alg. 2 & Trig.
Model Problems
Find the value of each summation
6
0
2k
k = 2(0 + 1 + 2 + 3 + 4 + 5 + 6)
6
0
2k
k
= 2(21) = 42
5
2
2
2n
n
= (2 – 2)2 + (3 – 2)2 + (4 – 2)2 + (5 – 2)2
= 02 + 12 + 22 + 32 = 14
31
1
j
j
j
1 1 2 1 3 1
0 1 2
1 2 3
1 2 3 12
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Aim: Summation Notation Course: Alg. 2 & Trig.
Model ProblemsRewrite using the summation symbol
2(1) + 2(2) + 2(3) + 2(4) + 2(5)5
1
2i
i
6
2
or 2( 1)i
i
4
0
1
3 3k k
1 1 1 1 1
3 6 9 12 15
1 2 3 4 5 61 2 3 4 5 6 6
1
n
n
n
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Aim: Summation Notation Course: Alg. 2 & Trig.
Regents Questions
2
2
0
The value of the expression 2 2 is
1) 12 2) 22 3) 24 4) 26
n
n
n
4
2
1Evaluate: 5
3 k
k
= 2
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Aim: Summation Notation Course: Alg. 2 & Trig.
In an arithmetic series, if a1 is the first term, n is the number of terms, an is the nth term, and d is the common difference, then Sn the sum of the arithmetic series, is given by the formulas:
The Sum of an Arithmetic Sequence: Series
)(2 1 naan
S
1[2 ( 1) ]2
nS a n d
or
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Aim: Summation Notation Course: Alg. 2 & Trig.
The nth Term of an Arithmetic Sequence
The nth term of an arithmetic sequence has the form
an = dn + c
where d is the common difference between consecutive terms of the sequence and
c = a1 – dAn alternative form of the nth term is
an = a1 + (n – 1)d
11 2
n
n ni
ndn c S a a
Summation and Arithmetic Series
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Aim: Summation Notation Course: Alg. 2 & Trig.
Regents Questions
Find the partial sum of the following arithmetic series
9
5
3 2k
k
= 115
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Aim: Summation Notation Course: Alg. 2 & Trig.
The Sum of a Finite Geometric Sequence
The sum of the finite geometric sequence a1, a1r2, a1r3, a1r4, . . . . a1rn - 1 . . . . with common ratio r 1 is given by
1
1
1
n
n
rS a
r
Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27, . . .
31
311
8
S 32802
6560
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Aim: Summation Notation Course: Alg. 2 & Trig.
Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27, . . .
The Sum of a Finite Geometric Sequence
r = ?3a1 a2 a3 a4 . . . . . an . . . .
a1 a1r a1r2 a1r3 . . . . a1rn - 1 . . . .
a1 + a1r + a1r2 +a1r3 + . . . + a1r8 - 1
81
11
( )n
n
a r
1
1
1
n
n
rS a
r
The sum of the finite geometric
sequence
=
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Aim: Summation Notation Course: Alg. 2 & Trig.
Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27, . . .
The Sum of a Finite Geometric Sequence
r = ?3
1 + 3 + 9 + 27 +. . .+ 1(3)7 = 3280
1 + 1(3) + 1(3)2 + 1(3)3 + . . . . = 3280
31
311
8
S 32802
6560
81
11
( )n
n
a r
8 81 1
11 1
( ) 1(3) 3280n n
n n
a r
1
1
1
n
n
rS a
r
=
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Aim: Summation Notation Course: Alg. 2 & Trig.
Find the sum of
10
0 2
110
i
i
Model Problem
10
0 2
110
i
i
1
1
1
n
n
rS a
r
a1 = 10; r = -1/2
How many terms in the series?
starting at i = 0, there are 11 terms
21
1
21
110
11
670.6
; n = 11
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Aim: Summation Notation Course: Alg. 2 & Trig.
Regents Questions
Find the sum of the following geometric series.
71
2
4(3)k
k
= 4368
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Aim: Summation Notation Course: Alg. 2 & Trig.
1
1
4(0.6)n
n
Find
= 4 + 4(0.6) + 4(0.6)2 + 4(0.6)3 + . .
The Sum of a Infinite Geometric Sequence
If |r| < 1, then the infinite geometric sequence a1, a1r2, a1r3, a1r4, . . . . a1rn - 1 . . . has the sum
11
1
n
na r
1
1
44(0.6) 10
1 0.6n
n
S
a1 = 4 and r = 0.6 - (|r| < 1)
1
1
aS
r
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Aim: Summation Notation Course: Alg. 2 & Trig.
Model Problem
Find the sum of 3 + 0.3 + 0.03 + 0.003 + . . .
r
araS n
n
1
11
11
1
1
aS
r
a1 = 3 and r = ?0.1
3 + 3(0.1) + 3(0.1)2 + 3(0.1)3 + . . .
3
1 (0.1)
3
0.9
13
3
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Aim: Summation Notation Course: Alg. 2 & Trig.
Model Problem
Express the series in sigma notation and find the sum.
54 + 18 + 6 + 2 + 2/3 + 2/9
728/9
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Aim: Summation Notation Course: Alg. 2 & Trig.
Model Problem
Find the sum of 12
1
4 0.3n
n
a1 = 4(0.3)1 = 1.2 n = 12r = 0.3
12
1 2 3 12
1
4 0.3 4 0.3 4 0.3 4 0.3 4 0.3n
n L
1
1
1
n
n
rS a
r
121 0.31.2
1 0.3nS
1.714