Arithmetic Sequences An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant amount The difference between consecutive terms is called the ID: 673512
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Slide1
Arithmetic Sequences
Dr. ShildneckSlide2
Arithmetic Sequences
An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant amount.The difference between consecutive terms is called the common difference of the sequence.
The common difference,
d
, can be found by subtracting
any two consecutive
terms (a
n
– an-1).
EXAMPLES
Arithmetic?
Common Difference (d)
1) 4, 7, 10, 13, 16, …
2) 8, 3, -2, -7, -12, …3) -2, -1/3, 4/3, 3, 14/3, …
Yes
Yes
Yes
+3
-5
+5/3Slide3
Arithmetic Functions
An arithmetic function with domain = {all positive integers} and range = {an} has a graph that consists of (only) points that lie on a straight line.
Thus, an arithmetic sequence can be thought of as a linear function whose domain is the set of all positive integers.
Furthermore, the common difference,
d
, is the rate of change of the function. Thus,
d
, is also the slope of the arithmetic/linear function.Slide4
Finding the Equation of an Arithmetic Function
Find the “slope” (the common difference)Compare each term value to its input (position) in the sequence.Write an equations (y = mx + b) that makes each input result in the appropriate term. (b is the adjustment that needs to be made)
[Example 1]
Write the equation for the sequence -3, 1, 5, 9, 13, 17, 21, 25, …Slide5
Developing a Formula for the nth Term
Find the equation for each term in an arithmetic sequence, based on
the given first term (a1) and common difference (d).
a
2
=
a
1
=
a
4
=
a
5 = a3 = a1a1 + da1 + d + da1 + d + d + d
a1 + d + d + d + d
= a1 + 2d = a
1 + 3d = a1 + 4d Slide6
Developing a Formula for the nth Term
Now looking at the pattern below, how might we write an equation for the n
th term (an) based on the first term (a1) and the common difference (d).
a
2
=
a
1
=
a
4
=
a
5 = a3 = a1a1 + d a1 + 2d a
1 + 3d a1 + 4d
Hint: Compare the “n” to the number of d’s required to get the n
th
term.
What do you notice about n and the number of d’s
you need for each term?Slide7
A Formula for the nth Term of an
Arithmetic Sequence
Given a first term (a1) and the common difference (d), The n
th
term (a
n
) of an arithmetic sequence can be found using the formula
a
n
= a
1
+ d(n – 1) Slide8
Examples
[Example 2] Find the 8th term of the arithmetic sequence whose first term is 4 and common difference is -7.Slide9
Examples
[Example 3] Find the nth term (formula) for the sequence.-2, 4, 10, 16, 22, 28, 34, 40, …Slide10
Examples
[Example 4] Find the formula for the nth term of the sequence.
Slide11
Examples
[Example 5] Find the nth term of of an arithmetic sequence whose first term
is 4 and fifth term in 40.Slide12
Examples
[Example 6] Find the formula for the nth term of an arithmetic sequence whose 7
th term is 86 and 18th term is 53.Slide13
ASSIGNMENT
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