Arithmetic Sequences as Linear Functions

8/6/2019 Arithmetic Sequences as Linear Functions

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Arithmetic Sequences as

Linear Functions


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Arithmetic SequencesA SEQUENCE is an ordered list of numbers.

Each number in a SEQUENCE is called a TERM.

In an ARITHMETIC SEQUENCE, each TERM is found by adding theSAME number to the previous TERM.

For example:

,,,,,,

The difference between each term is called the COMMONDIFFERENCE, .

In an ARITHMETIC SEQUENCE, the COMMON DIFFERENCE will be

the SAME for ALL terms in the sequence.


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Example 1 Identifying Arithmetic Sequences

Determine if,,,, is an ARITHMETIC

SEQUENCE:

It is fairly easy to see that the difference between

each term is constant we add 2 each time so

therefore, this is an ARITHMETIC SEQUENCE.


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You Try 1Determine if,,,, is an

ARITHMETIC SEQUENCE:

If you were correct, you should have found that the

common difference is CONSTANT we add 4 to each

term therefore, this is an ARITHMETIC SEQUENCE.


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Example 2Determine if

,

,

,

, is an ARITHMETIC SEQUENCE:

We need to see if the COMMON DIFFERENCE is constant so wehave:

+

=

+

=

=

+

=

Therefore, the COMMON DIFFERENCE does NOT remain constant,

so, we know that this is NOT an ARITHMETIC SEQUENCE.


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You Try 2Determine if,,,, is an ARITHMETIC

SEQUENCE:

If you were correct, you should have found that this

is NOT an ARITHMETIC SEQUENCE.


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Example 2 Finding the Next TermFind the next three terms of the arithmetic sequence ,,,,:

First, we need to find the common difference=

=

=

So the COMMON DIFFERENCE is

Now we can use this information to find the next three terms

+ =

+ =

+ =

So the next three terms in the ARITHMETIC SEQUENCE are ,,


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You Try 2Find the next three terms of the arithmetic sequence

.,.,.,.,:

If you were correct, you should have found that thenext three terms in the series are . , , and.


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More on Arithmetic SequencesEVERY term in an ARITHMETIC SEQUENCE can be expressed interms of the FIRST TERM, , and the COMMON DIFFERENCE, .

For example, the sequence , , , , can be thought of as

In general, the term of an ARITHMETIC SEQUENCE, with a firstterm of and common difference of, can be found by

= +

Where is a positive integer.


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Example 3 Finding the TermWrite an equation for the term of the arithmetic sequence,, , ,and then find the 9th term of the sequence. Next, graph the first five terms of the

sequence. Finally, determine which term of the sequence is 32:

First, we need to write the equation for the term of the arithmetic sequence ,,, ,

It is clear that the first term in the sequence, , is now, we need to first findthe common difference, .

It is clear that we are adding 4 to each term, therefore, = .

Now we can use this information to write our equation we have = +

Simply plug-in what we know and simplify

= +

= + =


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Example 3 - ContinuedNow we can use our equation to find the value of the 9th

term

=

Simply plug-in what we know and simplify

=

=

=

So the 9

th

term of the sequence is 20.


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Example 3 - ContinuedNow, we need to graph the first 5 terms of the sequence

We will think of as being our term #, and as being our term value

Term # Term Value

1 -12

2 -8

3 -44 0

5 4


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Example 3 - ContinuedFinally, we need to find out which term in the sequence has avalue of =

Again, we will use our formula

=

Now, plug-in what we know and simplify =

=

=

Therefore, the 12th term in the sequence has a value of 32.


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You Try 3Consider the arithmetic sequence ,,,,write an equation for the

term of the sequence andthen find the 15th term. Then determine which term in

the sequence has a value of.

If you were correct, you should have found that the

equation is = +

The 15th term in the sequence has a value of.

And the 10th term in the sequence has a value of.


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Homework #24

P.191, #s 8-23 all.

Arithmetic Sequences as Linear Functions

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