Arithmetic Sequences

Arithmetic Sequences

By Teri Salter

Multiple Choice

Which sequence is NOT an arithmetic sequence?

• -7, -13, -19, -25,...

• -10, -6, -2, 2,..

• 6, 8, 10, 12,...

• 3, 15, 75, 375,...

Multiple Choice

Which sequence is NOT an arithmetic sequence?

• 12, 18, 24, 30,...

• 10, 3, -4, -11,...

• 2, 4, 8, 16,...

• 16, 20, 24, 28,...

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, …

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

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

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

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

Fill in the Blanks

State the common difference:

Example 2: 21, 19, 17, 15, …

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

***What do the three dots (ellipsis) mean at the end of a sequence?***

• The sequence starts going in reverse order

• The sequence ends

• The sequence goes on forever

• The sequence increases by one from here on

Fill in the Blanks

Now, let's go backwards!!! Given the explicit formula, what is the first term?

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

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

TRUE or FALSE?

An arithmetic sequence is an example of a linear function.

• True, because having a common difference also means it has a constant rate of change

• False, because it's not shown on a graph