Arithmetic Sequences | Free Discrete Mathematics Lesson (4 College Credits) | UPI Study

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Arithmetic Sequences

Discrete Mathematics · Chapter 3: Sequences and SeriesSection 3.3 · lesson 19 of 877 min · free with a free account

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In this lesson

Start somewhere, add the same amount every step.

  • Define an arithmetic sequence and its common difference
  • Write recursive and explicit formulas
  • Find any term directly from its position
  • Identify an arithmetic pattern in raw data

Transcript

Everything said in the video, section by section (1,027 words). Read along, or search it.

Sequences and Series

Arithmetic Sequences

This section studies one of the most common patterns there is. Start at a fixed value, then add the same fixed amount every step. It's called an arithmetic sequence. We'll write its recursive and explicit formulas, find any term directly from its position, and learn to spot the pattern in raw data.

The Opening Problem

25 Results a Page

A web application shows search results twenty-five records at a time. It requests the first page starting at record offset zero, then offset twenty-five, then fifty, then seventy-five, and so on. Each page starts exactly twenty-five after the one before. The same pattern shows up constantly. A printer assigning job numbers one higher than the last. A parking garage's evenly spaced floor markers. A loan balance that drops by the same payment each month. This pattern turns out to be one of the most useful in the course. That is precisely because so much of it can be computed directly, without stepping through the terms in between.

Definition 3.3.1

The Common Difference

A sequence is arithmetic if there's a constant d, called the common difference, such that a sub n plus one minus a sub n equals d, for every n. Equivalently, each term is the previous term plus d. The common difference may be positive, negative, or zero. This states the definition directly. To get the next term, take the current term and add d. It's called the recursive formula, because computing a sub n requires already knowing a sub n minus one. And computing that requires the term before it, and so on, back to a one.

Worked Example

Page Offsets, Recursively

Here are the page offsets, computed recursively. The first offset is zero, and d is twenty-five. a two is zero plus twenty-five, which is twenty-five. a three is twenty-five plus twenty-five, which is fifty. a four is fifty plus twenty-five, which is seventy-five. And a five is one hundred. Each offset is produced from the one immediately before it. But for a distant term, that's expensive. Finding the fortieth offset this way means computing every term from the second to the thirty-ninth, first.

The Explicit Formula

Unwinding the Recursion

Unwinding the recursion by hand for the first few terms reveals a pattern that avoids this entirely. a two is a one plus d. a three is a one plus d, plus d, which is a one plus two d. And a four is a one plus three d. The coefficient of d is always one less than the term's index. Term two adds d once, term three adds it twice, and term four adds it three times. So the explicit formula is a sub n equals a one plus n minus one times d. A fully rigorous argument that this holds for every n uses mathematical induction, which comes in chapter four. The pattern is enough to work with now.

Jumping Straight There

Page 10's Starting Offset

With the explicit formula, page ten's starting offset comes directly. a ten equals zero, plus ten minus one, times twenty-five. That's nine times twenty-five, which is two hundred twenty-five. That's why a paginated API can jump straight to page ten's starting record. It evaluates one formula, instead of stepping through nine recursive updates.

A Negative Difference

Steady Battery Drain

The common difference doesn't have to be positive. A device under steady drain starts at one hundred percent charge, and drops by four percentage points every hour. So a one is one hundred, and d is minus four. The reading after fourteen more hours is a fifteen. That's one hundred, plus fourteen times minus four. One hundred minus fifty-six, which is forty-four percent. The sign of d decides the long-run behavior, in the sense of section three point two. Whenever d isn't zero, the terms grow without bound in magnitude. So an infinite arithmetic sequence with a nonzero difference is always unbounded, and always diverges. Of course, a real battery can't fall below zero percent. The abstract sequence and the physical process eventually part ways.

Figure 3.3.1

Always a Straight Line

This figure plots both running examples against n. On the left, the page offsets climb by twenty-five every step. On the right, the battery percentage falls by four every step. In both cases, the points fall on a single straight line. And the slope of that line is exactly the common difference. Positive for the offsets, negative for the battery.

Identifying From Data

Check Every Difference

Real data rarely arrives with a formula attached. It arrives as a list of numbers. To check whether a list is arithmetic, compute every consecutive difference, and confirm they're all exactly the same. Not just the first one or two. Take seven, twelve, seventeen, twenty-two, twenty-seven. Every difference is five. So it's arithmetic, with a one equal to seven and d equal to five. Now take three, six, nine, twelve, seventeen. The first three differences are all three, which suggests an arithmetic pattern. But the fourth difference is five. So the data isn't arithmetic, despite fitting the pattern for its first four terms. That's why the check must run through every pair. A sequence can look arithmetic for several terms, and still break the pattern on any later one.

Table 3.3.1

2 Strategies for One Sequence

This table compares the two formulas as computing strategies, using the page offsets. The recursive formula needs the immediately previous term. Finding the fortieth term costs thirty-nine additions. It's best suited to asking, what's the very next page's offset? The explicit formula needs only the first term, the common difference, and the target index. It costs one multiplication and one addition. It's best suited to asking, what's page forty's offset? Both describe the exact same sequence. The choice between them is about which question is being asked, not about which one is correct.

Takeaways

A Constant Step

To summarize. An arithmetic sequence has a constant common difference between consecutive terms. The recursive formula builds each term from the one before it. The explicit formula computes any term directly from its position. To identify arithmetic data, check that every consecutive difference is the same. And plotted against n, an arithmetic sequence always lies on a straight line, with slope d. Here's something to think about. Does the converse hold? If a sequence's points lie exactly on a straight line, must it be arithmetic? Could constant common difference and plots as a straight line ever come apart, or are they two descriptions of the same fact?

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FAQ

Questions students ask us

About Arithmetic Sequences

What is an arithmetic sequence and what is the common difference?
An arithmetic sequence is a sequence in which each term is the previous term plus the same constant d, called the common difference. In symbols, a_{n+1} - a_n = d for every n. The value of d can be positive, negative, or zero. For example, 0, 25, 50, 75 has d = 25.
What is the difference between the recursive and explicit formulas for an arithmetic sequence?
The recursive formula is a_n = a_{n-1} + d, which builds each term from the one before it, while the explicit formula a_n = a_1 + (n-1)d computes any term directly from its position. Recursion suits finding the very next term. The explicit formula suits a distant term, since it avoids calculating every term in between.
How do you find the 10th term of an arithmetic sequence without listing every term?
Use the explicit formula a_n = a_1 + (n-1)d with n = 10. For a sequence starting at a_1 = 0 with d = 25, the tenth term is a_{10} = 0 + 9 \cdot 25 = 225. The coefficient of d is one less than the term's index, so only one multiplication and one addition are needed.
How can you tell whether a list of numbers is an arithmetic sequence?
Compute every consecutive difference and confirm that all of them are exactly equal. Checking only the first few is a common mistake. The list 3, 6, 9, 12, 17 starts with differences of 3 but ends with 5, so it is not arithmetic. By contrast, 7, 12, 17, 22, 27 has d = 5 throughout.

About the Discrete Mathematics course

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Discrete Mathematics (Math 230) carries a NCCRS credit recommendation for 4 semester credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.
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