Arithmetic Series

Arithmetic Sequences

By Megan Cole

Arithmetic Sequence

An arithmetic sequence is an ordered list of numbers in which the difference between each pair of consecutive terms, or numbers in the list, is the same.

​Use the figures to complete the table. Plot the points given by your completed table. Describe the pattern of the y-values.

Exploration 1A

​Use the figures to complete the table. Plot the points given by your completed table. Describe the pattern of the y-values.

Exploration 1B

Draw

Exploration 1C: Fill in the table and draw a graph to represent the values.

Writing the Terms of Arithmetic Sequences

A sequence is an ordered list of numbers. Each number in a sequence is called a term. Each term a_n has a specific position n in the sequence.

Multiple Choice

EXAMPLE 1: Write the next three terms of the arithmetic sequence:

-7, -14, -21, -28, ...

• -33, -40, -47

• -40, -52, -64

• -35, -42, -49

• -30, -32, -34

Graphing Arithmetic Sequences

To graph a sequence, let a term's position number n in the sequence be the x-value. The term a_n is the corresponding y-value. Plot the ordered pairs (n, a_n).

​

Think about what we did at the beginning of the lesson​

Example 2

Graph the arithmetic sequence 4, 8, 12, 16, ....

​

​

​

​

What do you notice?

Open Ended

Example 3: Does the graph represent an arithmetic sequence? Explain.

​Writing Arithmetic Sequences as Functions

Because consecutive terms of an arithmetic sequence have a common difference, the sequence has a constant rate of change. So, the points represented by any arithmetic sequence lie on a line.

You can use the first term and the common difference to write a linear function that describes an arithmetic sequence. Let a₁ = 4 and d = 3.

Write an equation for the nth term of the arithmetic sequence 14, 11, 8, 5, ....

​

Then find a₅₀.

​

Example 4

Multiple Choice

Write an equation for the nth term of the arithmetic sequence.

4, 5, 6, 7, . . .

• $$a_n=4+1\left(n-1\right)$$  

• $$a_n=1+4\left(n-1\right)$$  

• $$a_n=4+2\left(n-1\right)$$  

• $$a_n=1+2\left(n-1\right)$$  

Multiple Choice

Find a₂₅ of the sequence:

4, 5, 6, 7, . . .

• 26

• 28

• 25

• 27

Multiple Choice

Write an equation for the nth term of the arithmetic sequence.

8, 16, 24, 32, . . .

• $$a_n=1+8\left(n-1\right)$$  

• $$a_n=8+4\left(n-1\right)$$  

• $$a_n=4+2\left(n-1\right)$$  

• $$a_n=8+8\left(n-1\right)$$  

Multiple Choice

Find a₂₅ of the sequence:

8, 16, 24, 32, . . .

• 192

• 208

• 200

• 216

Multiple Choice

Write an equation for the nth term of the arithmetic sequence.

1, 0, -1, -2, . . .

• $$a_n=1+\left(-1\right)\left(n-2\right)$$  

• $$a_n=1+\left(-1\right)\left(n-1\right)$$  

• $$a_n=1+1\left(n-1\right)$$  

• $$a_n=-2+1\left(n-1\right)$$  

Multiple Choice

Find a₂₅ of the sequence:

1, 0, -1, -2, . . .

• -23

• -22

• -24

• -25

​Writing Real-Life Functions

You can rewrite the equation for an arithmetic sequence with first term a₁ and common difference d in function notation by replacing a_n with f(n). The domain of the function is the set of positive integers.

​

f(n) = a₁ + (n - 1)d​

​Online bidding for a purse increases by $5 for each bid after the $60 initial bid.

​

a. Write a function that represents the arithmetic sequence.

​

​

b. Graph the function.

​

c. The winning bid is $105. How many bids were there?​

​

​

​

Example 5

Practice on your own

Use the IXL links in Google Classroom to do some practice.