Arithmetic Sequence Notes

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​Arithmetic Sequences

• Have your notebook out.

• Write down key vocabulary and concepts from the slides and video.

• Write down examples.
  
  

Notes:

Match

Match each term from the video with its meaning.

• sequence

• common difference

• arithmetic

• common ratio

• geometric

a list of numbers with a constant difference

the difference between consecutive terms in a sequence

a type of progression where the difference between terms is constant

the ratio between consecutive terms in a sequence

a progression where each term is multiplied by a constant factor

• a list of numbers with a constant difference

• the difference between consecutive terms in a sequence

• a type of progression where the difference between terms is constant

• the ratio between consecutive terms in a sequence

• a progression where each term is multiplied by a constant factor

Arithmetic Sequences (title)

Notes: Add to notebook.

Math Response

What is the first term of the sequence? a₁ =

Type answer here

Correct Answer: 7

Deg^°

Rad

Math Response

What is the fourth term of the sequence? a₄=

Type answer here

Correct Answer: 16

Deg^°

Rad

Math Response

What is the common difference?

(Term a_n - previous term (a_n-1)

Type answer here

Correct Answer: 3

Deg^°

Rad

Math Response

What is the 6th term of the sequence? a₆=

Type answer here

Correct Answer: 22

Deg^°

Rad

Open Ended

What did you do to find the 6th term? Be precise.

Notes Continued:

Arithmetic Sequences are sequences that follow a pattern based on adding/subtracting.

The number added to/subtracted from each term is a constant, fixed amount.

The difference between one term and the next is called the common difference.

Categorize

Options (20)

8

11

26

20

15

16.5

14

17

23

29

21

22.5

16

13

4

-2

20.5

18

15.5

3

Using a₁ and the common difference place the terms in the category that continues the sequence.

5, _, _,... common difference, d=3

12, _, _,.. common difference, d=1.5

19, _, _,... common difference, d= -3

23, _, _,... common difference, d= -2.5

So far, we have done problems like this...

Find the 8th term in the following sequence: a_n = 4, 6.5, 9, 11.5, 14,...

Easy right? We keep the sequence going and find that a₈= 18.5

But what if you had to find the 33rd term?

What if we needed to find the sum of those terms?

It would require a lot of adding and extra work, AND the chance of being off by a few numbers is very likely.

​

​

a_n= the term we are looking for
a_n-1= the previous term
d = common difference

Recursive Formula: a_n=a_n-1 + d

We have formulas to help find those terms that are further along the sequence.

Notes Continued:

​

​

a_n= the term we are looking for
a₁= the first term
d = common difference
n = the number of the term we are looking for

Explicit Formula: a_n=a₁ + d(n-1)

We have formulas to help find those terms that are further along the sequence.

Notes Continued:

​​Recursive

​​Explicit

Match

Determine whether the given sequence is arithmetic or not.

(Answer yes or no)

• 7, 14, 28, 56, 112

• 1, 4, 9, 16, 25

• 18, 27, 36, 45, 54

• 9, 5, 1, -3, -7

• 2, -4, 1, -6, 0

No

No

Yes

Yes

No

• No

• No

• Yes

• Yes

• No

Multiple Choice

The next two terms of the sequence 4, 7, 10, 13 are

• 16 and 19
• 15 and 19
• 16 and 18
• 15 and 18

Multiple Choice

Identify the common difference.
97, 86, 75, 64, ...

• 8
• 11
• -11
• -8

Multiple Choice

Given the sequence:
 25, 21, 17, 13,...
Write the explicit equation that models the sequence.

• a_n = 4n + 29
• a_n = -4n + 25
• a_n = -4n + 29
• a_n = 4n + 25

Multiple Choice

Create the explicit formula for the sequence:
2, 8, 14, ...
(Hint:  Write your formula and then simplify it.)

• a_n= 2 + 6n

• a_n= -6 + 6n

• a_n= 6n - 4

• a_n= 4 - 6n

Multiple Choice

Write a recursive rule for the sequence 17, 1, -15, -31 . . .

• a_n = a_n-1 + 16
• a_n = a_n-1 - 17
• a_n = a_n-1 - 15
• a_n = a_n-1 - 16

Multiple Choice

Write the recursive formula for the following sequence?
6, 4, 2, 0, . . .

• a_n = -2n + 8
• a₁ =6, a_n = a_n - 1 - 2
• a₁ = 6, a_n = a_n-1 + 2
• a₁ = 6, a_n = a_n-1 - 2

Multiple Choice

Find the 32nd term of this sequence.
9, 4, -1, -6, -11, ...

• 81
• -34
• -146
• -225

Multiple Choice

Find the 35th term in the arithmetic sequence: -10, -14, -18, -22, ...

• -216
• -148
• -146
• -150

Multiple Choice

Find a₂₆ in the arithmetic sequence: -15, -35, -55, -75, ...

• -515
• -490
• -535
• 460

Multiple Choice

Find the first term of the given sequence:
a₁₄=68
d=3

• 14
• 20
• 26
• 29

Multiple Choice

Write the RECURSIVE rule for the arithmetic sequence

-49, -35, -21, -7, ...

• a₁=-49;

  a_n = a_n-1 + 10

• a₁=-7;

  a_n = a_n-1 + 14

• a₁=-49;

  a_n = a_n-1 + 14

• a₁= -49;

  a_n = a_n-1 -14

Multiple Choice

Write the EXPLICIT rule for the arithmetic sequence

9, 13, 17, 21, ...

• a_n = -4n + 5

• a_n = -9n + 14

• a_n = 9n - 5

• a_n = 4n + 5

Multiple Choice

Identify the rule as recursive or explicit, then find the first 4 terms of the sequence.

a₁ = 100

a_n = a_n-1 - 8

• Explicit; 100, 92, 84, 76

• Recursive; 100, 92, 84, 76

• Explicit; 100, 108, 116, 124

• Recursive; 100, 108, 116, 124

​

Additional Video

Poll

How are you feeling about your introduction to arithmetic sequences?