MLF - Arithmetic Sequences (Lesson)

Arithmetic Sequences

A sequence is a special type of function. Each element in a sequence is called a term, these values would be considered the range. Each term is paired with a position number, and these values would be considered the domain. The domain of a sequence is a set of consecutive integers and typically starts at 0 or 1.

Position Number (domain)	$n$	1	2	3	4
Term of Sequence (range)	$f(n)$	2	4	6	8

From the table, you can see that the "third term in the sequence is 6" or $f(3) = 6$.

The explicit rule of a sequence is a rule that will allow you to determine any term in the sequence by using n, the position number. The explicit rule for the sequence above is $f(n) = 2n$.

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Arithmetic Sequences Practice

Answer this question set using the sequence shown above. 

 

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Using the Explicit Rule

Given $f(n) = 2n - 5$, we can plug in any value for n to find a solution, or given a value for f(n), you can solve for n.

1. If n is 0, $f(n)$ is -5.
2. If n is 2, $f(n)$ is -1.
3. If n is 3, $f(n)$ is 1.
4. If n is 5, $f(n)$ is 5.
5. If $f(n)$ is 3, n is 4.

A recursive rule for a sequence defines the terms of the sequence by relating it to one or more previous terms.

Watch this video to help you understand further:

 

An arithmetic sequence is a special type of sequence in which the difference between each term is constant. This difference is referred to as the common difference.

Watch this video to learn how to write recursive and explicit rules for arithmetic sequences.

 

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Common Difference and Sequence Practice

Determine if the sequences below are arithmetic by looking to see if there is a constant common difference between each of the numbers.   If the sequence is arithmetic, name the common difference, d.

1. {3, 7, 11, 15}
2. {1, 4, 9, 16}
3. {-8, -10, -12, -14}
4. {10, 7, 4, 10}
   
   

Write the first four terms of each sequence using a domain of consecutive integers starting with 1.

5. $f(n)=(n-1)^2$

6. $f(n) = -2n +7$

7. $f(1)=5$ and $f(n - 1) + 2$ for $n\ge2$.

8. $f(1) = 3$ and $f\left(n\right)=2\cdot f\left(n-1\right)-1$ for $n\ge2$.

Find the 10th term of the sequence assuming the domain is the set of consecutive integers starting with 1.

9. $f(n) = 5n + 7$

10.  $f(n) = n(n -1)$

Write an explicit rule for each sequence. Assume the domain for the function is the set of consecutive integers starting with 1.

11. 

$n$	$f(n)$
1	6
2	7
3	8
4	9

12. 

$n$	$f(n)$
1	3
2	6
3	9
4	12

13. 

$n$	$f(n)$
1	4
2	7
3	10
4	13

 

Write a recursive rule for each sequence.   Assume the domain for the function is the set of consecutive integers starting with 1.

14. 

$n$	$f(n)$
1	8
2	9
3	10
4	11

15. 

$n$	$f(n)$
1	2
2	4
3	8
4	16

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

 

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