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Arithmetic SequencesVocabulary
A set of numbers in a specific order is a SEQUENCE.
The numbers in the sequence are called TERMS
If the difference between terms is constant (the same), then it is called an ARITHMETIC SEQUENCE.
Arithmetic Sequences
In order to see if a sequence is an arithmetic sequence, find the difference between each term to see if it is the same. If it is the same than it is an arithmetic sequence.
Ex1: Is this an arithmetic sequence?
Answer: No this is not an arithmetic sequence because the difference between the terms is not the same (constant)
In order to see if a sequence is an arithmetic sequence, find the difference between each term to see if it is the same.
Ex1: Is this an arithmetic sequence?
Answer: Yes, this is an arithmetic sequence because the difference between terms is constant.
The difference between terms in an arithmetic sequence is called the COMMON DIFFERENCE.
What is the common difference for the following sequence?
Common Difference: +2
Writing Arithmetic Sequences
In words:
You can use the common difference of an arithmetic sequence to find the next term by adding it to the previous term
In symbols: (d is the common difference, a1 is the first term, a2 is the second term, a3 is the third term and so on.)
a1, a1+d, a2+d, …,
Ex: If the first term in an arithmetic sequence is 8 and the common difference is 4 find the next 3 terms.
Answer: 8, 12, 16, 20
+4
+4
+4
Writing Arithmetic Sequences
Example: The arithmetic sequence –8, –11, –14, –17, … represents the daily low temperature in ºF. Find the next three terms.
First find the common difference by subtracting successive terms.
The common difference is –3.
Second add -3 to the last term to get the next three terms. (remember adding -3 is the same as subtracting 3)
Try some on your own
The arithmetic sequence 58, 63, 68, 73, … represents the daily high temperature in ºF. Find the next three terms.
The arithmetic sequence 74, 67, 60, 53, … represents the amount of money tiffany owes her mother at the end of each week, find the next terms.
Find the next three terms for the following arithmetic sequence 9.5, 11.0, 12.5, 14.0,…
Exit-Slip
Describe what an arithmetic sequence is and how you find it.
What did you learn today? What are you still having trouble with?
Nth Term in a sequence
Term
symbol
Numbers
an+(n-1)(d)
8+(n-1)(3)
Each term in a sequence can be expressed in terms of the common difference d, and the first term in the sequence a1
Nth term in a sequence
The previous table leads to the equation to find the nth term in a sequence.
Example 1
The arithmetic sequence 1,10,19, 28, … is used to represent the total number of dollars Erin has in her bank account after her weekly allowance was added. Write an equation for the nth term in the sequence.
In order to write the equation we need a1 and d
What is a1 in this sequence? What is d in this sequence?
d= +9
We know a1=1, d=9 and
we know the formula an= a1 +(n-1)(d)
Plug in everything you know
an = a1 + (n –1)d Formula for the nth term
an = 1 + (n –1)(9) a1 = 1, d = 9
an = 1 + 9n – 9 Distributive Property
an = 9n – 8 Simplify.
Using the equation you got in example 1, Find the 12th term in the sequence
Wherever you see an n replace n with 12 in the equation.
an = 9n – 8 Equation for the nth term
a12 = 9(12) – 8 Replace n with 12.
a12 = 100 Simplify.
N
9n-8
an
Try one on your own
The arithmetic sequence 2, 7, 12, 17, 22, … represents the total number of pencils Claire has in her collection after she goes to her school store each week.
Write an equation for the nth term in the sequence.
Find the 12th term in the sequence using the equation you found.
Graph the first five terms (n, an)
Exit-Slip
How do you think arithmetic sequences relates to linear equations? What is similar about them? What is different?
What did you learn today? What do you still need help with?
Proportional and Non-Proportional
The graph always passes through the point (0,0)
Non-proportional relationship
A relationship in which a constant has to be added or subtracted
looks like y=kx+c
How do you know if a relationship is proportional or non-proportional?
In a proportional relationship is always the same or the graph passes through the origin (0,0).
In a non-proportional relationship is not always the same or does not pass through the origin.
Try some on your own (5 mins)
Determine if the following sequences are arithmetic sequences, if they are find the common difference.
2, 4, 8, 10, 12
-26, -22, -18, -14
Let’s play proportional or non-proportional.
Answer: Yes it is a proportional relationship because all the hours/miles are the same.
Answer: Yes it is a proportional relationship because all the hours/miles are the same.
Answer: No, not proportional because the graph does not pass through the origin
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