Sequences and Series

Algebra 2

Vocabulary

Sequence Series Term Domain Range Infinite Finite Summation (Sigma) Notation

Sequences and Series

Find the sum 1 + 2 + 3 + . . . + 9 + 10

Find the sum 1 + 2 + 3 + . . . + 98 + 99 + 100

Find the sum 1 + 2 + 3 + . . . + 999 + 1000

Find the sum 2 + 4 + 6 + . . . + 398 + 400

Sequences and Series

What is a sequence? What is a series? What is the difference? What is arithmetic, what is geometric? What is sigma (summation) notation? What formulas are important?

What is an Arithmetic Sequence?

An arithmetic sequence is when there is a common difference between consecutive terms.

The common difference is constant. It is written as d. Example: 2, 5, 8, 11, 14,… a1 = 2, a2 = 5, a3 = 8, a4 =11, a5 = 14 The common difference would be 3. The way you find “d” is by: 5 – 2 = 3, 8 – 5 = 3,

11 – 8 = 3, 14 – 11 = 3

Rule for an Arithmetic Sequence

The nth term of an arithmetic sequence with first term a1 and common difference “d” is:

1 ( 1)na a n d

Writing a Rule for the nth Term

Example: 2, 5, 8, 11, 14, … The common difference is 3 Use the equation : an= a1 + (n – 1)d a1= 2an= 2 + (n – 1)3an= 2 + 3n – 3 an= 3n – 1 That is your rule for the nth term

Johann Carl Friedrich Gauss

Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.

Johann Carl Friedrich Gauss

Another famous story has it that in primary school his teacher, J.G. Büttner, tried to occupy pupils by making them add a list of integers in arithmetic progression; as the story is most often told, these were the numbers from 1 to 100. The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher and his assistant Martin Bartels.

Gauss's presumed method was to realize that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050.

Find the Sum of a Finite Arithmetic Series

The expression formed by adding the terms of an arithmetic sequence is called an arithmetic series. The equation for the sum of a finite arithmetic series is:

1( )

2n

n

n a aS

Find the Sum of a Finite Arithmetic Series (cont.)

Example: 2 + 5 + 8 + 11 + 14 …

Find the sum of the first 20 terms. First off, you have to find a formula for the nth term. We did this on our

first example which came out to be : an= 3n – 1

Now you plug in 20 for n:

a20 = 3(20)-1=60-1=59

Now use to find the sum of first 20 terms

That is your answer

1( )

2n

n

n a aS

20

20(2 59)610

2S

Application #1

A well-drilling company charges $15 for drilling the first foot, $15.25 for the second foot, $15.50 for the third foot, and so on.

This means that it would cost $45.75 for the company to drill 3 feet.

How much would it cost for the company to drill a 100-foot well?

Application #2

The first row of a concert hall has 25 seats, and each row after that has one more seat than the row before it. There are 32 rows of seats.

Suppose each seat in rows 1 through 11 of the concert hall costs $24, each seat in rows 12 through 22 costs $18 and each seat in rows 23 through 32 costs $12.

How much money does the concert hall take in for a sold out event?

Geometric Sequences

With arithmetic sequences we added something each time

With geometric sequences we multiply by something each time

Geometric Sequences

Here are some geometric sequences:

3,6,12,24,...We multiplied by 2 each time.

1 1 1 11, , , , ,...2 4 8 16

We multiplied by –½ each time.

How do you find the ratios? Sometimes you can just look.

1,4,16,64,... The ratio is 4.

But usually you don’t get that lucky . . .

3 1 2 4, , , ,...2 3 27 243

OMG, how do you find the ratio here?

Geometric Sequences

Geometric Sequences have a common ratio. The common ratio is named r.

To find r we simply divide any term by its previous term.

32 4

1 2 3 1

... n

n

a aa ar

a a a a

Can you find the ratio from the sequence in the previous slide?

Find the ratio for the following sequence:

15 10 20,5, , ,...

2 3 9

Geometric Sequences

To find the general formula for a geometric sequence:

A geometric sequence is kind of like exponential growth or decay. You will multiply by a common ratio to find the next term.

11( )

nna a r

Find the general rule for the following:

32 6424,16, , ,...

3 9

Geometric Series

To find the sum of a finite geometric series, use the following formula:

1

1

1

n

n

rs a

r

Evaluate the following:

10

1

233

k

k

Geometric Infinite Series

Suppose everyday money was deposited into an account for you.

On the first day you received $100, on the next day you received $50, on the third day you received $25, and this pattern continued for the for the rest of your life and let’s assume you will live longer than Yoda (which I think is like more than 800 years)

How much money would be in your account?– What is happening to the amount of money being deposited?– About how much money is being deposited on the 30th day?– How do you think this will affect the balance?

Geometric Infinite Series

The sum of an infinite series can be found with the following formula:

1

1

aS

r

Evaluate the following:

1

1

212

3

n

n