Learning Resources
Lesson
Before we begin the work on Investigation 1 , there are some terms with which you need to be familiar. You may recall some or all of them from previous courses, but in the event you don't they are defined below.
Sequence: An ordered arrangement of numbers, symbols, or pictures in which each term follows another according to a rule.
Term: Each item in a sequence. We use subscripts to distinguish between the terms in a sequence. Thus the symbol t1 represents the first term in the sequence, t2 the second term, etc. tn will represent the nth term.
Consider the set of numbers {3 , 7, 11, 15, 19, 23}. This is an example of a sequence. The rule should be fairly obvious, viz. each term is found by adding 4 to the previous term. Some examples of the terms are: the first term is 3, so we can write t1 = 3; the fourth term is 15, so we can write t4 = 15; and the sixth term is 23, so we can write t6 = 23.
Finite sequence: A sequence is finite if it has a countable number of terms. In other words, the sequence ends. It is generally written as {t1, t2, t3, ...,tn} where tn represents the last or nth term.
Infinite sequence: A sequence is infinite if the number of its terms is unbounded. In other word, the sequence does not end. It is generally written as {t1, t2, t3, ...} where the three dots indicates the continuation of the sequence.
To see how a particular sequence might be developed lets use the problem of Investigation 1 on page 2 of your text. It states: "A contractor is building a chain-link fence containing 63 sections. If each section is a square made from four metal rods, and any two adjacent sections share one rod, how many rods are needed?"
One approach to solving this problem is shown in the interactive below. Work through it before proceeding to the next part of the lesson.
The formula developed above is called a recursive formula because each successive term is built upon the one preceding it. For this problem, knowing this formula or rule is not all that advantageous. We still have to continue finding each term until we reached the 63rd term. This would be a long tedious process.
Fortunately there is another way of developing a rule or formula that enables us to calculate our answer fairly quickly. Once you learn this method it can be applied to any sequence of this type.
The same fencing problem is shown in the step by step interactive below. However, a different formula is produced by considering what is to be added to the first term in order to get any term in the sequence.
Using the above formula, the 63rd term of the sequence (the number of rods needed for the fence) is:
t63 = 4 + (63-1) x 3 = 4 + 186 = 190
If we look at the above sequence {4, 7, 10, 13, 16, 19 ,..., 187, 190} and make a new sequence of differences by finding the difference between successive terms (t2 - t1, t3 - t2, etc.) we get {3 , 3 , 3 , 3 , ...,3 , 3}. The differences are all the same. When that happens we say there is a common difference and it is usually designated as d. Also, any sequence that has a common difference is called an arithmetic sequence.
You should now be able to generalize the formula we derived for the nth term of the fence problem to any sequence which has a common difference.
If a sequence is arithmetic (i.e. has a common difference ), and has first term t1 and common difference d, then the nth term can be found using the formula:
tn = t1 + (n - 1)d
Note that in the above formula, the domain (the values of the independent variable n) is the set of natural numbers {1 , 2, 3, 4, ...}. This gives discrete values for the range (the values of the dependent variable tn). The graph of a sequence is thus a set of discrete points. It is incorrect to draw the graph of the sequence as a continuous curve.
However, if the function for generating a sequence is graphed using the set of all real numbers for the domain, the graph becomes a continuous curve containing the points of the sequence.
An example of the application of sequences is found below.
Example
Determine whether { 2, 6, 10, 14, ...} is arithmetic. If so, write the nth term as a function of the first term and use it to find the value of the 35th term.
Solution
t2 - t1 = 6 - 2 = 4
t3 - t2 = 10 - 6 = 4
t4 - t3 = 14 - 10 = 4
There is a common difference of 4 ( d = 4 ) and the first term is 2 ( t1 = 2 ). Using the formula developed on the previous page, the formula or function rule for the nth term of this sequence is:
tn = t1 + (n-1)d
tn = 2 + (n - 1) x 4
tn = 2 + 4n - 4 (removing the brackets)
tn = 4n - 2 ( simplifying)
Thus the 35th term is: t35 = 4(35) - 2 = 140 - 2 = 138
Activity
- If you have completed the lesson you need not do Investigation 1 on page 2 of your text.
- Complete Investigation Questions 1 to 6 on pages 3 & 4.
- Do the CYU Questions 7 to 15 on pages 4 to 6.
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, click on the Test Yourself button at the top of the page for a quick quiz on this lesson.
Test Yourself
Use the sequence { 2 , 3.5 , 5 , 6.5 , ...} to answer the following questions:
- Is the sequence arithmetic? Why?
- Write the function rule for the nth term of the sequence and use it to find the 25th term.
- Using domain of all the real numbers, graph the function used to generate the sequence . How does the slope of the graph of the function compare to the common difference?