Math II

UNIT QUESTION: How is a geometric sequence like an exponential function?Standard: MM2A2, MM2A3

Today’s Question:How do you recognize and write rules for number patterns?Standard: MM2A3d

Section 4.7

Sequences&

Series

SequenceA sequence is a set of numbers

in a specific order

Infinite sequence

Finite sequence

,...,...,,,, 4321 naaaaa

naaaaa ,...,,,, 4321

Sequences – sets of numbers

Notation:

represents the formula for finding terms

term numberna

n

Examples:

If 2 3, find the first 5 terms.na n

term. 20th the find ,13 If nan

th4

nd32

is the notation for the 4 term

is the notation for the 32 term

a

a

Ex.1 Find the first four terms of the sequence

23 nan

12)1(31 a42 a

73 a

104 a

First term

Second term

Third term

Fourth term

Ex. 2

12

)1(

n

an

n

Find the first four terms of the sequence

Writing Rules for Sequences

We can calculate as many terms as we want as long as we know the rule or equation for an.

Example:

3, 5, 7, 9, ___ , ___,……. _____ .

an = 2n + 1

an = a1 + (n-1)d

Writing Rules for Sequences

Try these!!!

3, 6, 9, 12, ___ , ___,……. _____ .

1/1, 1/3, 1/5, 1/7, ___ , ___,……. _____ .

an = 3n, an = 1/(2n-1)

Series – the sum of a certain number of terms of a sequence

n

1iia :Notation Sigma

“Add up the terms in the sequence beginning at term number 1 and going through term number “n”.

Start

Stop

Formula

4

1i

5 1. i-

5

1i21 1 2. i

7

3i

3. i

5 1 5 2 5 3 5 4 50

25

25

2

6

i 1

4. 3

Infinite Sequence ,...,...,,,, 4321 iaaaaa

Infinite Series

1

4321 ......i

ii aaaaaa

Finite Series

n

iin aaaaaa

14321 ...

or nth Partial Sum

Series Notation Practice

• Workbook Page 155 #19-24

Arithmetic Series

Arithmetic Series

• The sum of the terms in a sequence are called a series.

• There are two methods used to find arithmetic series:

Formula

Sigma Notation

Arithmetic SeriesSequence Sum Average Avg. x n

2, 5, 8, 11, 14

40 8 40

1, 8, 15, 22

46 23/2 46

-1,1, 3 3 1 3

What’s another way to get the average without adding all the numbers and dividing by n?

Arithmetic Series

• a1 = first term

• n = number of terms

• an = value of the nth term

In many problems, you will first have to use the arithmetic sequence formula.

1

2n

n

a aS n

Arithmetic Series• Find the sum of the arithmetic series.

18

1

4 2i

i

648

Arithmetic Series• Find the sum of the arithmetic series.

100

1

2i

i

10,100

Arithmetic Series

• Find a formula for the partial sum of this series:

1+3+5+....+(2n-5)+(2n-3)+(2n-1)

1

2n

n

a aS n

1 2 1

2n

nS n

2nS n