Arithmetic Sequences and Series

GOOD NEWS!

You know almost everything you need to know about this class of sequences already!

An arithmetic sequence is nothing more than a linear function with the specific domain of the natural numbers. The outputs of the function create the terms of the sequence. Let's look at how the concepts relate.

Take the point-slope linear form . Let's translate the elements here into sequence lingo.

Recall:

    1. The terms of a sequence are the outputs of some function , so . (Look back at the introductory lesson.)
    2. The domain of a sequence is usually the natural numbers. Let's use n for them. So, x = n in our formula.
    3. The value m is the slope in a linear function. In the sequence world as we go from term to term, we find that the change in input is always 1 while the change in output never changes. It is common to all consecutive pairs of terms. In the sequence world the slope is exactly the same as the common difference, d. Then m = d.
    4. The first term is always labeled . It is the ordered pair (1, ). We'll use it for the (h, k) point in the point-slope form.

Putting them all together we have a rule for creating nth term formula:

Examples:

For the sequence 11, 15, 19, 23, . . .

    1. Is this an arithmetic sequence?
    2. Write the nth term formula.
    3. Calculate the value of the 30th term.

Typical applications look like these:

  1. A field house has a section where the seating can be arranged so the first row has 11 seats, the second row has 15 seats, the third row has 19 seats and so on. If there is sufficient space for 30 rows in the section, how many seats are in the last row of the section?

    2. We answered this above. There are 127 seats in the section.

  2. A company began doing business four years ago. Its profits for the last 4 years have been $11 million, $15 million, $19, million and $23 million. If the pattern continues, what is the expected profit in 26 years?

    3. Hum.... Same calculation, but the answer is $127 million.

  3. A production line is improving its efficiency through training and experience. If the number of items produce in the first four days of a month are 11, 15, 19, and 23, respectively. Project the number of items produced by the end of a 30 day month if the pattern continues.

    4. Been there, done that. The answer is 127 items of production on day 30.

Finding the Sum of an Arithmetic Series

The arithmetic family of sequences and their associated series one of the easiest to work with. We can create a rule for calculating sums of the terms by just looking at how the calculations come out for a few simple examples.

    1. I've calculated the sum for the series . You might notice that when you average the first and last term and multiply by the number of terms you get the same total. Maybe it's a coincidence, but let's look at another linear series.
    2. For the series , again the sum is the average of the first and last term multiplied by the number of terms.
    3. For the series , it works again! And, this arithmetic series is decreasing!

Now we can take more time thinking about the rule, but the formula for the sum of a finite arithmetic series, often symbolized by is

Just remember, "average the first and last and multiply by the number of terms."

Now let's dispense with one piece of silliness, the sum of an infinite arithmetic series. It cannot work by any formula. The average of the first and last terms is undefined since we are using an infinite value in the process. Or, look at the sequence of terms, they are going off to some infinity. Either way, it is silly.

However, there are other times when it is completely fair to ask about the sum of an infinite series. You'll see them in the next lesson about geometric sequences and series.

 

Let's apply the formulas to these examples:

A field house has a section where the seating can be arranged so the first row has 11 seats, the second row has 15 seats, the third row has 19 seats and so on. If there is sufficient space for 30 rows in the section, how many seats are in the section?

Notice the change in the question? To do this problem we need the first and last terms. The complete calculation is to the right. We can seat 2070 people in the section.

 

 

 

A company began doing business four years ago. Its profits for the last 4 years have been $32 million, $38 million, $42 million and $48 million. If the pattern continues, what is the expected profit total profit in the first ten years.

The common difference is 6 million dollars per year and the first term is $32 million. Using those values, the answer is $590 million.

 

 

 

 

A production line is improving its efficiency through training and experience. If the number of items produce in the first four days of a month are 13, 15, 17, and 19, respectively. Project the total number of items produced by the end of a 30 day month if the pattern continues.

The common difference is 2 units per day and the first term is 13 items. Using those values, the answer is 1260 items.