aim: what is this symbol it’s greek to me!

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


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.


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.


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


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


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


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


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


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


Regents Questions

Find the partial sum of the following arithmetic series

9

5

3 2k

k

= 115


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




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

=




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

=


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


Regents Questions

Find the sum of the following geometric series.

71

2

4(3)k

k

= 4368


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


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


Model Problem

Express the series in sigma notation and find the sum.

54 + 18 + 6 + 2 + 2/3 + 2/9

728/9


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

aim: what is this symbol it’s greek to me!

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