Bellwork Extend each pattern 3 more terms, then

describe in words how each term relates to the one previous.

a) -3, 0, 3, 6, …

b) 2, 4, 8, …

c)

9, 12, 15

Each term is the previous term plus 3

16, 32, 64

Each term is the previous term multiplied by 2

Each term is the previous term multiplied by 1/2

End in Mind Title your notes page Sequences- Day 1 Put a sub-header “End-in-mind” Copy the problems below and use any

strategies/resources to solve.

A) A car whose original value was $25,000 decreases in value by $250 per month. How long will it take before the car’s value falls below $20,000?

B) A car whose original value was $25,000 decreases in value by 5% per month. After 1 year, how much will the car be worth?

Sequences: A set of values arranged in a specific order, a pattern

Recursive Process: Used to describe a pattern or sequence by describing how to get from one term to the next.

Explicit Expression: Used to describe a pattern or sequence so that any term in the sequence can be found.

Vocabulary

The value added each time is called the "common difference"The common difference could also be negative:

Example:25, 23, 21, 19, 17, 15, ...This common difference is −2

(Vocabulary Continued)

We call the common difference ‘d’

The value multiplied each time is called the "common ratio"

The common ratio could also be a fraction:

Example:48, 24, 12, 6, 3, 1.5, ...This common ratio is 1/2

(Vocabulary Continued)

We call the common ratio ‘r’

Use a Recursive Process to determine the 10th y-value

+4

+4

+4

Each y-value is the previous one plus 4.

So the 10

th term

is…

41

Arithmetic

Use an Explicit Expression to find the 10th term

4+1

4 +4+1

4 +4 +4+1

4 +4 +4 +4+1

Each y-value is the x-valuetimes 4, plus 1.

So… y=4x+1

and the 10th term is…y = 4(10)+1 = 41

Still Arithmeti

c

x4, +1

Remember: We added 4 each time…

To determine the Explicit Expression for Arithmetic Sequences

1) Determine what you are adding each time. (the common difference)

x(term

)

y

1 -2

2 2

3 6

4 10

Added 4 each time, so we start off with y=4x…

y=4x would lead to…y=4(1)=4

2) Adjust to fit the pattern

Our first term is -2, NOT 4.So we need to subtract 6.

y=4x-6

For each of the following arithmetic sequences. (a) Determine the 8th term using the Recursive process. (b) Determine the 20th term using an Explicit Expression.

1. 2. 3. x(term

)

y

1 3

2 6

3 9

4 12

x(term

)

y

1 1

2 3

3 5

4 7

x(term

)

y

1 -2

2 1

3 4

4 7

I will come around and check these as you complete them.

Sequences Day 1- I.C. Practice

Use a Recursive Process to determine the 10th y-value

x3

x3

x3

Each y-value is the previous one times 3.

So the 10

th term

is…

78,732

Geometric

4

12

36

108

Use an Explicit Expression to find the 10th term

4

4x3

4x3x3

4x3x3x3

Each y-value is the first y-value

times 3 to the x-value minus 1.

So… y=4 x 3x-1

and the 10th term is…y = 4 x 310-1=4 x 39=78,732

Still Geometric

4 x 33

Remember: We multiplied by 3 each time…

4

12

36

108

To determine the Explicit Expression for Geometric Sequences

1) Determine what you are multiplying by each time. (the common ratio)

x(term

)

y

1 2

2 -4

3 8

4 -16

Multiply each time by -2, so we start off with y=(-2)x-1

y=(-2)x-1 would lead to…y=(-2)1-1=(-2)0=1

2) Adjust to fit the pattern

Our first term is 2, NOT 1.So we need to multiply by 2.

y=2(-2)x-1

For each of the following geometric sequences. (a) Determine the 6th term using the Recursive process. (b) Determine the 12th term using an Explicit Expression.

1. 2. 3. x(term

)

y

1 4,096

2 2,048

3 1,024

4 512

x(term

)

y

1 -2

2 -8

3 -32

4 -128

x(term

)

y

1 5

2 -5

3 5

4 -5

I will come around and check these as you complete them.

Sequences Day 1- I.C. Practice

Ticket Outa) Come up with your own example of an

arithmetic sequence and state the explicit expression that corresponds with it.

b) Come up with your own example of a geometric sequence and state the explicit expression that corresponds with it.

Notation: d= common difference (what is added

to each term of an arithmetic sequence) r= common ratio (what is multiplied to

each term of a geometric sequence) n= What term number you are looking

at (4th term, 10th term… nth term)

1st term 2nd term 3rd term 4th term 5th term nth term

a1 a2 a3 a4 a5 an