Arithmetic Sequences Lesson

Sequences

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

example: 5, 10, 15, 20, 25,…a_n

Arithmetic Sequence

An arithmetic sequence is a sequence where you add a constant number to find the next term, similar to the constant rate of change in a linear function.

In an arithmetic sequence we only add to find the next number (term). This number can be positive or negative.

Term

In a sequence each number is called a term.

The dots at the end mean the sequence continues forever.

a₁ is the first term

a₂ is the second term

a₃ is the third term and so on...

a_n is the nth term (we will find the equation for the nth term in the next lesson)

Common Difference

The number you add/subtract to find the next term in a sequence is called the common difference.

Subtract two terms to find the common difference. d is used to define the common difference

Ready to try some examples?

Remember in an arithmetic sequence, each number is found by adding a common difference, which can be positive or negative.

Multiple Choice

Identify the common difference.

10, 20, 30, 40, ...

• 5

• 10

• 15

• 20

Multiple Choice

The next two terms of the sequence 12,17,22,27 are

• 28 and 33

• 29 and 34

• 32 and 37

• 30 and 35

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

Find the common difference: 29, 21, 13, 5, ...

Remember, the common difference is the difference between the numbers.

• d = -10

• d = -9

• d = -8

• d = -7

Multiple Choice

Is the sequence arithmetic: 37, 31, 25, 19, ...

• Yes
• No

Multiple Select

Which sequences are arithmetic?

(hint: select 2)

• 2, 4, 6, 8, 10 ...

• 5, 10, 20, 40, 80, 100 ...

• 1, 3, 7, 9, 15, 20 ...

• 4, 8, 12, 16, 20, 24 ...

Multiple Choice

If the first term of an arithmetic sequence is 5 and the common difference is 6, what are the first five terms?

• 5, 11, 17, 23, 29

• 6, 11, 16, 21, 26

• 5, 10, 15, 20, 25

• 5, 11, 15, 20, 26

Multiple Choice

Which sequence has a common difference of 4?

• 4, 6, 8, 10, 12...

• 1, 5, 9, 13, 17....

• 4, 4, 4, 4, 4...

• 4, 14, 24, 34, 44...

Multiple Choice

We will start with the explicit formula. Given the following sequence, state the first term and common difference.

Example 1:          4, 7, 10, 13, 16, …

• $$a_1\ =\ 3,\ \ \ d\ =\ 4$$    

•   $$a_1\ =\ 16,\ \ \ d\ =\ 3$$  

•   $$a_1\ =\ 4,\ \ \ d\ =\ 3$$  

•   $$a_1\ =\ 16,\ \ \ d\ =\ 13$$  

Multiple Choice

Now, write the explicit formula for the given sequence.

Example 1:          4, 7, 10, 13, 16, …

• $$a_n\ =\ 4\ +\ \left(n\ -\ 1\right)\left(3\right)$$  

• $$a_n\ =\ 3\ +\ \left(n\ -\ 1\right)\left(4\right)$$  

• $$a_n\ =\ 7\ +\ \left(n\ -\ 1\right)\left(3\right)$$  

• $$a_n\ =\ 16\ +\ \left(n\ -\ 1\right)\left(4\right)$$  

Multiple Choice

Write an explicit rule to model the following sequence:

Example 2: 21, 19, 17, 15, …

• $$a_n\ =\ 21\ +\left(n\ -\ 1\right)\left(2\right)$$  

• $$a_n\ =\ -21\ +\left(n\ -\ 1\right)\left(2\right)$$  

• $$a_n\ =\ -21\ +\left(n\ -\ 1\right)\left(-2\right)$$  

• $$a_n\ =\ 21\ +\left(n\ -\ 1\right)\left(-2\right)$$  

Multiple Choice

Write and explicit rule to model the following sequence:

Example 3:        -12, -7, -2, …

• $$a_n\ =\ 5\ +\ \left(n\ -1\right)\left(-12\right)$$  

• $$a_n\ =\ -12\ +\ \left(n\ -1\right)\left(5\right)$$  

• $$a_n\ =\ -5\ +\ \left(n\ -1\right)\left(12\right)$$  

• $$a_n\ =\ 12\ +\ \left(n\ -1\right)\left(-5\right)$$  

Multiple Choice

Find the 26th term in the arithmetic sequence: 20, 26, 32, 38, ...

• 182
• 176
• 170
• -118

Multiple Choice

Now, let's go backwards!!! Given the explicit formula, list the first five terms in order.

$$a_n\ =\ 3\ +\ \left(n\ -\ 1\right)\left(2\right)$$  

• 2, 5, 8, 11, 14, ...

• 3, 1, -1, -3, -5, ...

• 3, 6, 9, 12, 15, ...

• 3, 5, 7, 9, 11, ...

Now, let's simplify the explicit formula even further

​  a_n = 4 + (n - 1)(-2) **What do you see that we can do to this equation?

Some text here about the topic of discussion

Now, our equation is a_n = -2n + 6

​  a_n = -2n + 6 **How does this resemble linear equation in slope- intercept form?

​

​

y = mx + b​ Slope-Intercept Form

Some text here about the topic of discussion

Arithmetic Sequences Model Linear Functions!!!

​  a_n = -2n + 6 **The common difference is the same as the slope!

​

​

y = mx + b​ **The y-intercept is the difference of the first term

and the common difference... or b = a₁ - d

Some text here about the topic of discussion

Multiple Choice

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

• a_n = 6n + 2

• a_n = 6n - 6

• a_n = 6n - 4

• a_n = -6n + 4

Multiple Choice

Find the 25th term of the sequence if a₁ = 5 and d = 0.5

• 18
• 17
• 15
• 14

Multiple Choice

Find the explicit formula:
-3, -33, -63, -93, ...

• a_n = -3 - 30(n -1)
• a_n = -3 + 30(n -1)
• a_n = -30 - 3(n -1)
• a_n = -30 + 3(n -1)

Multiple Choice

Find the explicit formula:
-23, -16, -9, -2, ...

• a_n = -23 + 7(n -1)
• a_n = -23 - 7(n -1)
• a_n = 7 - 23(n -1)
• a_n = 7 + 23(n -1)

Multiple Choice

Find the term named in the problem
14, 22, 30, 38, ... a₂₂

• 190
• 182
• 169
• 127

Multiple Choice

Find the term named in the problem
14, 214, 414, 614, ... a₃₀

• 5724
• 5814
• 5784
• 5614

Multiple Choice

Given the arithmetic sequence
a_n = 14 + (n-1)9 
which equation is equivalent

• a_n = 9n-5
• a_n = 9n+5
• a_n = 5n +9
• a_n = 9 + 14(n-1)