Practice: Arithmetic and Geometric Series | Free Discrete Mathematics Lesson (4 College Credits) | UPI Study

UPI Study › Courses › Discrete Mathematics › Free lessons › Lesson 3.8 Free with a free account

Practice: Arithmetic and Geometric Series

Discrete Mathematics · Chapter 3: Sequences and SeriesSection 3.8 · lesson 24 of 878 min · free with a free account

← Sum of Geometric SeriesNext lesson: Principles of Mathematical Induction →

Earn 4 college credits for Discrete MathematicsRecommended for credit by NCCRS · Math 230 · self-paced · one payment, lifetime access
Enroll in Discrete Mathematics · $250 →Download the syllabus (PDF)

In this lesson

Diagnose the series first - then reach for the formula.

  • Test differences and ratios to identify a series
  • Recognize real-world cue phrases for each type
  • Solve applied arithmetic and geometric problems
  • Combine totals when a growth pattern changes

Transcript

Everything said in the video, section by section (1,194 words). Read along, or search it.

Sequences and Series

Practice: Arithmetic and Geometric Series

Sections three point six and three point seven derived the two closed-form sum formulas. No new formula appears in this section. What it adds is practice with the skill that has to come first. Correctly recognizing, from a real-world description, which kind of series a scenario actually represents.

The Opening Problem

2 Columns, 2 Kinds of Growth

A growing logistics startup's planning spreadsheet tracks two kinds of growth side by side. The delivery fleet grows by a fixed number of vans every quarter, since the leasing budget adds the same increment each time. The number of app users grows by a fixed percentage every quarter, since each new group of users refers a proportional number of new signups. Both columns grow, and both eventually need a running total. But plugging the wrong closed-form formula into the wrong column doesn't raise an error. It just quietly produces the wrong number.

Quick Reference

The 2 Formulas, From 3.6 and 3.7

Here are the two formulas, exactly as derived earlier. An arithmetic series has a constant difference d. Its sum is n over two, times the first term plus the last. A geometric series has a constant ratio r. Its sum is a one, times one minus r to the n, over one minus r, for r not equal to one. Both types produce a growing sequence, with a positive difference or a ratio above one. So growth alone isn't a reliable signal. The reliable test looks at how consecutive terms relate to each other.

The Diagnostic Test

Differences, or Ratios?

Given at least three consecutive terms, here's the test. Compute the differences, a two minus a one, and a three minus a two. If they're equal, the sequence is arithmetic, and d is that common difference. Compute the ratios, a two over a one, and a three over a two. If they're equal, the sequence is geometric, and r is that common ratio. If neither pattern holds, the sequence is neither purely arithmetic nor purely geometric, and neither formula applies directly. This isn't a separate rule to memorize. Each definition guarantees its own pattern. The test just checks, from the outside, which definition the data actually satisfies.

Figure 3.8.1

A Quick Visual Check

This figure shows two sequences, both starting at five, and both increasing. The arithmetic one, with difference three, traces a straight line. The geometric one, with ratio one point four, curves upward. The shape is often the fastest visual sanity check for which formula applies. But the difference and ratio test is the reliable one.

Worked Example 3.8.1

2 Sequences, 2 Diagnoses

A warehouse records shelving units bought over three months. Forty, forty-six, fifty-two. A marketing report records an account's followers. Five hundred, six hundred fifty, eight hundred forty-five. For the shelving, the differences are six and six. Equal, so it's arithmetic, with first term forty and difference six. The ratios, one point one five and about one point one three, aren't equal. For the followers, the differences are one hundred fifty and one hundred ninety-five. Not equal, so it isn't arithmetic. The ratios are one point three and one point three. Equal, so it's geometric, with first term five hundred and ratio one point three. The two look superficially similar, three increasing numbers each. Only the test, not a glance, reveals which formula each one needs.

Table 3.8.1

Diagnostic Checklist

This checklist consolidates the cues. An arithmetic series has a constant difference. Its cue phrases are, increases by a fixed amount, or adds X more units each period. Typical scenarios are fixed-increment loan payments, flat fee or staffing increases, and linear capacity build-out. A geometric series has a constant ratio. Its cue phrases are, increases by a fixed percentage, grows X percent each period, doubles, or compounds. Typical scenarios are compound interest, population or user growth at a constant rate, and viral referral growth.

Worked Application

A Loan Repayment Schedule

A borrower repays a small business loan in ten monthly installments. The first payment is two hundred dollars, and each one after that increases by a fixed forty dollars. Diagnose first. Each payment adds a fixed dollar amount, so this is an arithmetic series. The first term is two hundred, d is forty, and n is ten. Now solve. The last payment is two hundred plus nine times forty, which is five hundred sixty. The total is ten over two, times two hundred plus five hundred sixty. That's five times seven hundred sixty, or three thousand eight hundred dollars repaid over the ten months.

Worked Application

Compounding Population Growth

A city planning department models a new development corridor, using illustrative figures. Twelve hundred new residents move in during year one, growing by a fixed six percent each year, over six years. Diagnose. The count grows by a fixed percentage, not a fixed head count. So it's geometric, with first term twelve hundred, ratio one point zero six, and six terms. Build up the power one multiplication at a time. One point zero six to the sixth is about one point four one eight five. Apply the geometric sum formula. Twelve hundred, times one minus one point four one eight five, over one minus one point zero six. That's about eight thousand three hundred seventy point four. So about eight thousand three hundred seventy new residents, rounded to a whole person.

Mixed Practice

Diagnose, Then Solve

Two short practice scenarios, each needing the full workflow. A recycling program collects three, three and a half, four, and four and a half tons in its first four weeks. The differences are all one half, so it's arithmetic. The tenth week collects seven and a half tons. So ten weeks total five times ten and a half, or fifty-two and a half tons. A new podcast gets two hundred, two hundred sixty, and three hundred thirty-eight downloads in its first three weeks. The differences, sixty and seventy-eight, aren't equal. But the ratios are both one point three. So it's geometric. One point three to the sixth is about four point eight two six eight, and six weeks total about two thousand five hundred fifty-one downloads. The habit worth keeping is the one in every example. Test differences and ratios first. Only then reach for a formula, never the other way around.

When the Pattern Changes

Split, Then Combine

Suppose the corridor grows by a fixed two hundred residents a year for its first three years, while infrastructure is built. Then, once transit opens, it switches to compounding six percent annual growth from year four. No single arithmetic or geometric formula can compute the total across all six years directly. But you can compute each piece with its own closed form, and then add them. That's exactly what the splitting rule from section three point five allows. Two separately computed totals combine into one grand total.

Takeaways

Test Before You Compute

To summarize. Before applying either sum formula, test whether consecutive terms differ by a constant amount or a constant factor. Applying the wrong formula silently produces the wrong total. Real-world language is a good first cue. A fixed amount signals arithmetic, and a fixed percentage signals geometric. The difference and ratio test confirms it. And a scenario whose pattern changes partway through can be split, summed piece by piece, and combined. One last example. A tutoring center records signups of twenty, twenty-four, twenty-eight, and thirty-two. A colleague says it must be geometric, because it's growing. But the differences are all four, so it's arithmetic. Growth alone never distinguishes the two types. Only the test does.

About Discrete Mathematics at UPI Study

Discrete Mathematics (Math 230) is a self-paced online course of 87 video lessons in 13 chapters, from logic and proof to counting, probability, graph theory and trees. It carries a NCCRS credit recommendation for 4 semester credits.

Every lesson transcript is free on pages like this one, and the videos are free with a free UPI Study account. Enrolling adds a practice quiz after each lesson, the graded work and final exam that earn the credit, lifetime access, and an official transcript on completion. UPI Study has helped over 130,000 students with college credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Enroll now → Course details Download the syllabus Every course with PRO →

Why take it here

Why students take Discrete Mathematics with UPI Study

The math behind computer scienceDiscrete Mathematics is the mathematics of computer science: logic, proofs, sets, counting, probability and graphs underpin algorithms, data structures, cryptography and databases. Completing it now lets you qualify for those courses at your university.
Credit at a fraction of the priceThe same 4 credits can cost $750 or more at a US college. Here it is one payment, self-paced, with lifetime access.
Finish your degree soonerTransfer credits reduce the courses you still owe, so you can graduate earlier or lighten a heavy semester.
Strengthen an applicationA completed college-level course on an official transcript shows admissions and scholarship committees you can do the work.
Study on your own scheduleNo start dates, no deadlines. Watch the lessons, practice, and take the final when you are ready.
130,000+students helped with college creditsACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Where the credit goes

Universities that work with NCCRS

Discrete Mathematics carries a NCCRS credit recommendation. These universities work with NCCRS; some are UPI Study's signed partners. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

Thomas Edison State UniversityThomas Edison State UniversitySigned partnerUniversity of the PeopleUniversity of the PeopleSigned partnerCharter Oak State CollegeCharter Oak State CollegeSigned partnerStrayer UniversityStrayer UniversitySigned partnerColumbia Southern UniversityColumbia Southern UniversitySigned partnerIndiana TechIndiana TechSigned partnerPenn StatePenn StateWorks with NCCRSSouthern New Hampshire UniversitySouthern New Hampshire UniversityWorks with NCCRSSUNYSUNYWorks with NCCRS +many more cooperating universitiesFind my college →

Find my college → Not listed? Ask Athena in the corner of this page, or book a 15-minute call.

FAQ

Questions students ask us

About Practice: Arithmetic and Geometric Series

How do you tell whether a series is arithmetic or geometric?
Test consecutive terms: compute the differences a_2-a_1 and a_3-a_2, then the ratios a_2/a_1 and a_3/a_2. Equal differences mean arithmetic, and equal ratios mean geometric. For 500, 650, 845, the differences are 150 and 195, but both ratios are 1.3, so it is geometric. Growth alone cannot decide, because both types can increase.
What cue phrases in a word problem signal an arithmetic or a geometric series?
Phrases like "increases by a fixed amount" or "adds X more units each period" signal an arithmetic series, as in fixed-increment loan payments. Phrases like "grows X percent each period," "doubles," or "compounds" signal a geometric series, as in compound interest or user growth. Confirm the cue with the difference and ratio test before choosing a formula.
How do you find the total repaid on a loan with 10 monthly payments that start at 200 dollars and rise by 40 dollars each month?
The total is 3800 dollars. A fixed 40-dollar increase is a constant difference, so the series is arithmetic with a_1=200, d=40, and n=10. The last payment is 200+9\cdot 40=560. Then S_{10}=\frac{10}{2}(200+560)=3800. Diagnosing the series type first tells you which sum formula to use.
How do you total a growth pattern that changes partway through?
Split the pattern into pieces, compute each piece with its own closed-form formula, and add the totals. For example, if a population grows by a fixed number of residents for three years and then compounds at 6 percent for three more, sum the arithmetic part and the geometric part separately. The geometric part uses the year-4 value as its first term and has three terms.

About the Discrete Mathematics course

How many college credits is Discrete Mathematics?
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.
How much does Discrete Mathematics cost and what is included?
$250, paid once, with lifetime access: all 87 video lessons, a practice quiz after each, the graded work and final exam, and an official transcript when you finish. Students who want several courses choose UPI Study PRO instead.
Which universities accept UPI Study credits?
UPI Study has signed partner agreements with universities such as Thomas Edison State University, University of the People and Charter Oak State College, and works with universities such as Penn State, SNHU and SUNY. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities. Use "Find my college" on this page to check yours.
How long does Discrete Mathematics take?
It is self-paced with no deadlines. The 87 lessons total about 13 hours of video; most students add reading and practice time and finish in a few weeks to a few months, on their own schedule.
How are the exams done?
Quizzes and assignments are online in the course. The final exam is taken online with proctoring; a person on the UPI Study team checks assignments, usually within two days.
How do my credits reach my college?
When the course shows 100% complete, request the official transcript from your dashboard. It is sent to the college you name, and you can send it to more than one. Check your college first with "Find my college" on this page, or ask Athena.

Ready for the full course? Earn 4 credits in Discrete Mathematics.

87 video lessons, a quiz after each one, and the graded work that earns the credit. Recommended for credit by NCCRS.

Enroll in Discrete Mathematics →Download the syllabusEvery course with PRO →
Discrete Mathematics
4 credits · NCCRSEnroll →