11.1 An Introduction to Sequences &amp; Series

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11.1 An Introduction to Sequences &amp; Series. p. 651. What is a sequence? What is the difference between finite and infinite?. Sequence :. A list of ordered numbers separated by commas. Each number in the list is called a term . For Example: Sequence 1 Sequence 2

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### 11.1 An Introduction to Sequences & Series

p. 651

What is a sequence?
• What is the difference between finite and infinite?
Sequence:
• A list of ordered numbers separated by commas.
• Each number in the list is called a term.
• For Example:

Sequence 1Sequence 2

2,4,6,8,10 2,4,6,8,10,…

Term 1, 2, 3, 4, 5 Term 1, 2, 3, 4, 5

Domain – relative position of each term (1,2,3,4,5) Usually begins with position 1 unless otherwise stated.

Range – the actual terms of the sequence (2,4,6,8,10)

Sequence 1Sequence 2

2,4,6,8,10 2,4,6,8,10,…

A sequence can be finite or infinite.

The sequence has a last term or final term.

(such as seq. 1)

The sequence continues without stopping.

(such as seq. 2)

Both sequences have a general rule: an = 2n where n is the term # and an is the nth term.

The general rule can also be written in function notation: f(n) = 2n

Write the first 6 terms of an=5-n.

a1=5-1=4

a2=5-2=3

a3=5-3=2

a4=5-4=1

a5=5-5=0

a6=5-6=-1

4,3,2,1,0,-1

Write the first 6 terms of an=2n.

a1=21=2

a2=22=4

a3=23=8

a4=24=16

a5=25=32

a6=26=64

2,4,8,16,32,64

Examples:
The seq. can be written as:

Or, an=2/(5n)

The seq. can be written as:

2(1)+1, 2(2)+1, 2(3)+1, 2(4)+1,…

Or, an=2n+1

Examples: Write a rule for the nth term.
Example: write a rule for the nth term.
• 2,6,12,20,…
• Can be written as:

1(2), 2(3), 3(4), 4(5),…

Or, an=n(n+1)

Graphing a Sequence
• Think of a sequence as ordered pairs for graphing. (n , an)
• For example: 3,6,9,12,15

would be the ordered pairs (1,3), (2,6), (3,9), (4,12), (5,15) graphed like points in a scatter plot

* Sometimes it helps to find the rule first when you are not given every term in a finite sequence.

Term #

Actual term

Graphing

n

1

2

3

4

a

3

6

9

12

What is a sequence?

A collections of objects that is ordered so that there is a 1st, 2nd, 3rd,… member.

• What is the differencebetween finite and infinite?

Finite means there is a last term. Infinite means the sequence continues without stopping.

Assignment:

p. 655

9-29 all,

skip 23

Sequences and Series Day 2
• What is a series?
• How do you know the difference between a sequence and a series?
• What is sigma notation?
• How do you write a series with summation notation?
• Name 3 formulas for special series.
Series
• The sum of the terms in a sequence.
• Can be finite or infinite
• For Example:

Finite Seq.Infinite Seq.

2,4,6,8,10 2,4,6,8,10,…

Finite SeriesInfinite Series

2+4+6+8+10 2+4+6+8+10+…

Summation Notation
• Also called sigma notation

(sigma is a Greek letter Σ meaning “sum”)

The series 2+4+6+8+10 can be written as:

i is called the index of summation

(it’s just like the n used earlier).

Sometimes you will see an n or k here instead of i.

“the sum from i=1 to 5 of 2i”

i goes from 1

to 5.

Summation Notation for an Infinite Series
• Summation notation for the infinite series:

2+4+6+8+10+… would be written as:

Because the series is infinite, you must use i from 1 to infinity (∞) instead of stopping at the 5th term like before.

a. 4+8+12+…+100

Notice the series can be written as:

4(1)+4(2)+4(3)+…+4(25)

Or 4(i) where i goes from 1 to 25.

Notice the series can be written as:

Examples: Write each series in summation notation.
Example: Find the sum of the series.
• k goes from 5 to 10.
• (52+1)+(62+1)+(72+1)+(82+1)+(92+1)+(102+1)

= 26+37+50+65+82+101

= 361

Example: Find the sum.
• Use the 3rd shortcut!
What is a series?

A series occurs when the terms of a sequence are added.

• How do you know the difference between a sequence and a series?

The plus signs

• What is sigma notation?

• How do you write a series with summation notation?

Use the sigma notation with the pattern rule.

• Name 3 formulas for special series.
Assignment:

p. 655

30-61 every 3rd problem