Learning Resources
Lesson
Read the purpose and procedure for this investigation and generate the requested sequence with the first and second level differences as indicated. To re-create the problem, if you are having difficulty, you and a partner might want to "act out" the problem, recording the results for each situation; look for patterns or start with the simple case of 2 outlets, 2 people, 3 outlets, 2 people, and so on, listing the choices.
The ideas of sequences of differences is extended from the first-level for arithmetic sequences to finding second-level differences and continuing to third-level, fourth-level and so on.
A sequence 
will have first level differences represented by
Using this new sequence, we subtract the adjacent terms creating 
, the second-level differences. This process can be continued to find higher level differences.
Keeping these differences in mind, complete the assigned Investigation Questions under Activities to explore the quadratic sequence before continuing.
Recall that a quadratic function can be expressed as 
where 
and a and b are coefficients and c is a constant. These are second-degree functions since the greatest power in the expression is 2. During this lesson, you should have discovered a relationship between the coefficient a and the second-level differences in a sequence generated by a quadratic function (i.e. a quadratic sequence). Investigation Question 21 leads you to this relationship.
Complete the investigation questions from the Activities Section before continuing with this lesson.
When generating a quadratic sequence from the form 
, it can be completed by substituting the term number and calculating each individually or it can be completed through the use of technology through the recursive feature of the TI-83. Then by using the List editor we can generate the sequences of differences.
Click here for a demonstration of how the first and second level differences can be found on the TI-83 for the first ten terms of the sequence generated by the function 
.
Example
Generate a quadratic sequence with at least ten terms using the equation 
. Find its second level differences and state the relationship between -2 and the second-level differences.
Solution
Use the technology or substitute the values 1 to 10 inclusive into the equation and calculate each individually to find the following sequence:
{-1, -5, -13, -25, -41, -61, -85, -113, -145, -181}
The first-level differences will be
The second-level differences are
The second-level differences is -4 which is twice the coefficient -2. "For a quadratic function , the second-level differences is 2a"
Activity
Investigation Questions p.8. Complete 17, 18, 19, 20, and 21
Think About p.10
Challenge Yourself p.10 and p.11
Check Your Understanding p. 9. Complete 22 to 33 inclusive
When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.
After you do the assigned activities, continue on to the Test Yourself section for a quick quiz on this lesson.
Test Yourself
Write a Journal entry explaining, using sequences of differences, how an arithmetic sequence differs from a quadratic sequence
In your Journal, explain what is meant by a power sequence.
Determine the value of a for the quadratic function that generates the sequence {-4, 5, 18, 35, 56, ...}.
Click here for sample solution.