Arithmetic Sequences and Series

Section 9-2

2

Objectives

• Use sequence notation to find terms of any sequence

• Use summation notation to write sums

• Use factorial notation

• Find sums of infinite series

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9.1 Sequences & SeriesSEQUENCE: A list that is

ordered so that it has a 1st term, a 2nd term, a 3rd term and so on.

example: 1, 5, 9, 13, 17, …a1 = 1; a2

= 5; a3 = 9, etc.

The nth term is denoted by: an

The nth term is used to GENERALIZE about other terms.

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The three dots mean that this sequence is INFINITE.

example: 1, 5, 9, 13, 17, …

example: 2, -9, 28, -65, 126

This is a FINITE sequence.

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Given a “rule” for a sequence,

find the sequence.

2

4

14 1 7

2f

211

2nf x

2

1

1 11 1

2 2f 2

2

12 1 1

2f

2

3

1 73 1

2 2f

2

5

1 235 1

2 2f

1 7 23,1, ,7,

2 2 2

EXAMPLE 1:

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Write the first four terms of the sequence whose nth term is given by:

12

)1(

n

n

na

17

19

15

13

1

4

3

2

1

a

a

a

a

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Recursion Formula

• Defines the nth term of a sequence as a function of the previous term.

235 11 nn aaanda

51 a

23 12 aa

172)5(32 a

532)17(33 a

1612)53(34 a

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Find the first four terms

523 11 nn aaanda

123

59

27

11

4

3

2

1

a

a

a

a

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Factorial Notationn! = n(n – 1)(n – 2)…1

Special case: 0! = 1

8! 8 7 6 5 4 3 2 1 8 Math/prb/4/enter

= 40,320

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A sequence is arithmetic if the differences between consecutive terms are the same.

4, 9, 14, 19, 24, . . .

9 – 4 = 5

14 – 9 = 5

19 – 14 = 5

24 – 19 = 5

arithmetic sequence

The common difference, d, is 5.

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Example: Find the first five terms of the sequence and determine if it is arithmetic.

an = 1 + (n – 1)4

This is an arithmetic sequence.

d = 4

a1 = 1 + (1 – 1)4 = 1 + 0 = 1

a2 = 1 + (2 – 1)4 = 1 + 4 = 5

a3 = 1 + (3 – 1)4 = 1 + 8 = 9

a4 = 1 + (4 – 1)4 = 1 + 12 = 13

a5 = 1 + (5 – 1)4 = 1 + 16 = 17

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The nth term of an arithmetic sequence has the form

an = dn + c

where d is the common difference and c = a1 – d.

2, 8, 14, 20, 26, . . . .

d = 8 – 2 = 6

a1 = 2 c = 2 – 6 = – 4

The nth term is: an = 6n – 4.

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a1 – d =

Example: Find the formula for the nth term of an arithmetic sequence whose common difference is 4 and whose first term is 15. Find the first five terms of the sequence.

an = dn + c

= 4n + 11

15,

d = 4

a1 = 15 19, 23, 27, 31.

The first five terms are

15 – 4 = 11

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The sum of a finite arithmetic sequence with n terms is given by

1( ).2n nnS a a

5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = ?

( )501 2755 )0 5(552nS

n = 10

a1 = 5 a10 = 50

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The sum of the first n terms of an infinite sequence is called the nth partial sum.

1( )2n nnS a a

( )190 25(184) 4602

50 6 0nS

a1 = – 6

an = dn + c = 4n – 10

Example: Find the 50th partial sum of the arithmetic sequence – 6, – 2, 2, 6, . . .

d = 4 c = a1 – d = – 10

a50 = 4(50) – 10 = 190

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• The sum of the first n terms of a sequence is represented by the summation notation

• Where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.

Summation Notation

n

n

ii aaaaaa

...43211

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Consider the infinite sequence a1, a2, a3, . . ., ai, . . ..

1. The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence.

1

n

ii

a

a1 + a2 + a3 + . . . + an

2. The sum of all the terms of the infinite sequence is called an infinite series.

1i

i

a

a1 + a2 + a3 + . . . + ai + . . .

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Example

5

1

)25(i

i• Expand and evaluate the sum:

Solution:

5

53113

)25(5

1

i

i

19

100

1

2i

n

Example: Find the partial sum.

2( ) 2( ) 2( ) 2( )1 2 3 100 2 4 6 200

a1 a100

100 1 10010( ) 2( )02 0

2 20nS a a

50(202) 10,100

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Homework

• WS 13-4