- I. Summary and Preliminary Notes
- II. Calculations and Accounts
- III. Creativity
- IV. Historical (or other) Context
- V. Group Work
- VI. Personalization
- VII. Answering Difficult Students’ Questions
- VIII. Using Other Sources
- IX. Safety and Ethics
- X. Bibliography

In this article, I would like to present you, as a math teacher, 13 specific examples of how Chat GPT can improve and facilitate your work.

- Access to ChatGPT, preferably version 4, as the article is based on this version

- High school
- College

Need a step-by-step solution to a problem? No problem, just ask Chat GPT.

In an infinite arithmetic sequence (a_n), where n >= 1, the sum of the first eleven terms is 0, and the arithmetic mean of the third, fifth, and seventh terms is (-2). The terms (a_2, a_4, a_k) form a geometric sequence in the given order. Determine k.

Chat GPT has incredible capabilities for interpreting mathematical formulas. It understands (based on context) LaTeX, notations like “a_n”, “x^2”, and much more.

Additionally, it usually repeats the formulas you used, giving you the opportunity to correct any misinterpretations.

Use this to save a lot of time and energy on converting different ways of writing formulas.

ChatGPT handles solving such an example excellently:

You | |

Solve the problem step by step: In an infinite arithmetic sequence (a_n), where n >= 1, the sum of the first eleven terms is 0, and the arithmetic mean of the third, fifth, and seventh terms is (-2). The terms (a_2, a_4, a_k) form a geometric sequence in the given order. Determine k. Copy | |

ChatGPT | |

Let’s solve the problem step by step. ## Data:1. The sum of the first eleven terms of the arithmetic sequence is 0. ## Step 1: Determining the basic parameters of the arithmetic sequenceThe arithmetic sequence can be described by the general formula: ## Sum of eleven terms:The sum S_{11} of the first eleven terms is 0: ## Arithmetic mean of the third, fifth, and seventh term: \frac{a_3 + a_5 + a_7}{3} = -2 Arithmetic mean: Thus: ## Step 2: Determining the terms a_2, a_4 and a_kCalculate a_2 and a_4: ## Step 3: Determining kThe terms a_2, a_4, a_k form a geometric sequence, which means: Substitute: Find the value of k, which is the term equal to -2: ## Answer:k = 5 | |

Creating examples for students to solve requires a lot of energy from us, teachers. Chat GPT can do it in a few seconds.

You | |

Create a typical example of finding the limit of a sequence using conjugation for a college-level problem. Copy | |

ChatGPT | |

## Typical Example of Finding the Limit of a Sequence Using ConjugationFind the limit of the sequence a_n = \frac{\sqrt{n^2 + 3n} – n}{2n + 1} as n approaches infinity. (…) ## Answer:\lim_{n \to \infty} \frac{\sqrt{n^2 + 3n} – n}{2n + 1} = \frac{3}{4} | |

Chat GPT can design an entire test for you in a few seconds.

You | |

Create test questions on arithmetic and geometric sequences for high school level. The test should have 5 questions. The last question should be a word problem. Copy | |

ChatGPT | |

Of course, here are five questions for a high school test on arithmetic and geometric sequences: ## Question 1: Arithmetic SequenceGiven the arithmetic sequence a_n, where the first term a_1 = 5 and the common difference d = 3. a) Write the general formula for this sequence. ## Question 2: Sum of an Arithmetic SequenceCalculate the sum of the first twenty terms of the arithmetic sequence where the first term a_1 = 7 and the common difference d = 2. ## Question 3: Geometric SequenceGiven the geometric sequence b_n, where the first term b_1 = 4 and the ratio q = 2. a) Write the general formula for this sequence. ## Question 4: Sum of a Geometric SequenceCalculate the sum of the first five terms of the geometric sequence where the first term b_1 = 3 and the ratio q = \frac{1}{2}. ## Question 5: Word ProblemAdam saves a fixed amount of money each month, which forms an arithmetic sequence. The first deposit was 50 PLN, and each subsequent deposit was 10 PLN higher than the previous one. a) What was Adam’s eighth deposit? ## Answers and Solutions:## Question 1: Arithmetic Sequencea) General formula for the arithmetic sequence: a_n = a_1 + (n-1)d b) Tenth term: ## Question 2: Sum of an Arithmetic SequenceSum of the first n terms of an arithmetic sequence: S_n = \frac{n}{2} (2a_1 + (n-1)d) ## Question 3: Geometric Sequencea) General formula for the geometric sequence: b_n = b_1 \cdot q^{n-1} b) Eighth term: ## Question 4: Sum of a Geometric SequenceSum of the first n terms of a geometric sequence: S_n = b_1 \frac{1-q^n}{1-q} ## Question 5: Word Problema) Adam’s eighth deposit: b) Total savings after eight months: | |

Just ask Chat GPT 🙂 Of course, don’t follow them too strictly; modify them according to your needs.

You | |

Create a 45-minute high school lesson plan. The topic of the lesson is monotonicity of sequences. Copy | |

ChatGPT | |

## 45-Minute High School Lesson Plan
## Lesson Objectives:– Understand the concept of sequence monotonicity ## Preparation:– Whiteboard and markers ## Lesson Plan:## 1. | |

Chat GPT can easily create a multiple-choice test for you from scratch.

You | |

Create a multiple-choice test with 10 questions and A/B/C/D answers. The test should cover mathematical sequences, their definition, applications, types (arithmetic and geometric sequences), monotonicity, convergence (with definition). Copy | |

ChatGPT | |

## Multiple Choice Test on Mathematical Sequences## Question 1: What is an arithmetic sequence?A) A sequence where each term is the product of the previous term and a constant
## Question 2: Which of the following formulas describes the general term of an arithmetic sequence?A) a_n = a_1 \cdot q^{n-1}
## Question 3: What is a geometric sequence?A) A sequence where each term is the sum of the previous term and a constant
## Question 4: Which of the following formulas describes the general term of a geometric sequence?A) a_n = a_1 + (n-1)d
## Question 5: What does it mean for a sequence to be monotonic?A) A sequence where each term is positive
## Question 6: Which of the following expressions describes an increasing sequence?A) a_n > a_{n+1} for every n
## Question 7: Which of the following expressions describes a decreasing sequence?A) a_n \geq a_{n+1} for every n
## Question 8: What is the value of the sum of the first n terms of an arithmetic sequence S_n?A) S_n = a_1 \cdot q^n
## Question 9: What does it mean for a sequence to be convergent?A) Every term of the sequence is the same
## Question 10: Find the limit of the sequence a_n = \frac{1}{n} as n approaches infinity.A) 1
| |

If you can interact with ChatGPT in front of your students (e.g., by sharing your laptop screen on a projector), use this opportunity. Let ChatGPT impersonate a historical figure, and have the students ask questions.

First, you create the character in ChatGPT, which the students do not need to see.

You | |

From now on, you are a historical figure – Fibonacci (Leonardo of Pisa). Immerse yourself in the character, pretend to be Italian, and use some Italian words. Students will ask you questions about the Fibonacci sequence (which you discovered) and other topics. Copy | |

ChatGPT | |

(…) | |

Now you can give the floor to your students:

You | |

Hello Copy | |

ChatGPT | |

Ciao! Welcome, my young students! I am Leonardo di Pisa, known as Fibonacci. How can I help you today? Do you have questions about my famous sequence, its applications, or perhaps other mathematical curiosità? Please, feel free to ask! | |

You | |

What do you look like? Can we see your picture? Copy | |

ChatGPT | |

Ah, my dear students, unfortunately, in my time there were no cameras, quindi non ci sono foto di me. However, I can describe how I might have looked according to contemporary depictions. I am depicted as a middle-aged man with a long beard, dressed in traditional attire of those times, usually in a long tunic and cloak. I might have worn a cap or a scarf on my head. My appearance reflected a scholar of the medieval period, immerso nelle sue studi e scritture. Of course, your imagination may vary, but the most important are my mathematical works and contributions to the development of science, not how I looked. I am happy to answer any questions about my work and achievements! | |

You | |

What are your achievements? Copy | |

ChatGPT | |

Ah, sono felice che lo chiediate! My greatest achievements are related to my work on introducing the numerical system we now call the ## 1. “Liber Abaci” (The Book of Calculation)– In 1202, I wrote “Liber Abaci” or “The Book of Calculation”. In this work, I introduced the Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the decimal system, which are much easier to use than Roman numerals. This revolutionized mathematics in Europe! ## 2. The Fibonacci Sequence– My famous sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, \ldots, was first described in “Liber Abaci” in the context of a problem related to rabbit reproduction. ## 3. The Golden Ratio– The Fibonacci sequence is closely related to the golden ratio (\phi), which is approximately 1.618. The further along the sequence, the closer the ratio of successive terms approaches the golden ratio. ## 4. Solving Commercial Problems– In my book, I also described how to use new methods to solve commercial problems, such as calculating profits and losses, converting currencies, and calculating percentages. These methods were very practical and useful for merchants and traders. ## 5. Introduction of Algorithms– In my works, one can find early examples of algorithms, which are the foundation of modern computer science. I described methods for solving equations and calculating square roots, which can be considered precursors to today’s algorithms. These are just some of my achievements. I am glad that my works have had such a significant impact on the development of mathematics and science in general. Prego, if you have more questions, feel free to ask! | |

You | |

In what times did you live? Copy | |

ChatGPT | |

Ah, i tempi erano molto diversi da oggi! I lived at the turn of the 12th and 13th centuries, specifically between about 1170 and 1250. It was the Middle Ages, a time when Europe was undergoing many changes, both social and cultural. ## My Time and Place:– ## My Works:– ## The Middle Ages:– Life in my times was full of challenges but also incredible opportunities. I am glad that I could contribute to the development of mathematics and influence future generations. If you have other questions, feel free to ask! | |

You | |

Show us on the map where Pisa is. Copy | |

ChatGPT | |

Ah, Pisa! It’s a beautiful city located in Tuscany, in central Italy. Let me show you on the map where exactly Pisa is. ![Map of Italy with Pisa marked](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Italy_adm_location_map.svg/640px-Italy_adm_location_map.svg.png) Pisa is marked in the Tuscany region, near the coast of the Tyrrhenian Sea. It is, of course, famous for its iconic If you have more questions about Pisa or anything else, feel free to ask! | |

You | |

Have you ever been to Poland? Copy | |

ChatGPT | |

Ah, la Polonia! Unfortunately, I did not have the opportunity to visit Poland in my time. My travels were mainly around the Mediterranean basin, where I had the chance to learn Arabic and Indian mathematics and bring these ideas to Europe. I traveled from my hometown Pisa to places such as: However, I know that Poland has a rich history and culture, as well as its own achievements in mathematics and science. Perhaps other questions about my work, my life, or the Fibonacci sequence? I am happy to answer! | |

You | |

Did you have a girlfriend? Copy | |

ChatGPT | |

Ah, young students, amore is always an interesting question! My private life is not well documented, as in my time, the focus was mainly on intellectual and scientific achievements. There are no historical records indicating that I had a girlfriend or wife. In my work, I focused mainly on mathematics and introducing new ideas to Europe. Traveling, researching, and writing were my main activities. These were times when scientific work required many sacrifices and could occupy a scholar’s entire life. However, I am sure that love and relationships were important in my time, just as they are today. Are there any other questions about mathematics, my work, or life in the Middle Ages? I am happy to answer! | |

You | |

Could you practice mathematics in the Middle Ages? Weren’t you burned at the stake? Copy | |

ChatGPT | |

Ah, capisco le vostre preoccupazioni! Indeed, the Middle Ages were a time of intense change, including religious and social tensions. However, science and mathematics had their place and were valued, especially for practical applications in trade, construction, and navigation. ## Some facts about science in the Middle Ages:1. 2. 3. ## Was I in danger?My work focused mainly on the practical aspects of mathematics, such as the decimal system, fractions, and commercial calculations, which were very useful and well-received. Mathematics was a tool that helped in everyday life, not seen as a threat. Of course, there were times when knowledge and innovative ideas could be controversial, but my work aimed primarily at improving daily life and trade. Perhaps that’s why I managed to avoid major conflicts. If you have more questions about my work or life in the Middle Ages, I am happy to answer! | |

You | |

Mr. Fibonacci, we have one last question and please – be absolutely honest with us. Did you invent the Fibonacci sequence?Copy | |

ChatGPT | |

Ah, carissimi studenti, I understand your doubts and I’m glad you asked! Allow me to answer honestly. The sequence we call the ## History of the sequence:– ## My Contribution:– My greatest achievement was introducing this sequence to Europe and showing its practical application in “Liber Abaci.” Thanks to my work, this sequence became known and widely used. ## Did I invent the sequence?– No, I did not invent this sequence first, but I was certainly the first to describe it in the context of European mathematics and introduce it to broader use. I hope this explanation is sufficient. I am proud to have contributed to the development of mathematics, and I am glad my works are still valued and studied. If you have any more questions, I am at your disposal! | |

You can ask ChatGPT to come up with anything, including games for students.

You | |

I am a high school math teacher. I would like to engage my students in a game related to mathematical sequences during a 45-minute class. My class has 20 students, and I want to divide them into 5 groups. What game ideas related to mathematical sequences do you have for my students?Copy | |

ChatGPT | |

Understood, here are some game ideas related to mathematical sequences that can engage students during a 45-minute class: ## Game 1: Mathematical Bingo
## Game 2: Mathematical Relay
## Game 3: Guess the Sequence
## Game 4: Sequence Puzzles
## Game 5: Mathematical Scrabble
I hope these suggestions help students better understand mathematical sequences and make the lesson engaging and interactive! | |

You | |

Come up with topics for students to work on in groups related to mathematical sequences. I want to divide my class of 20 students into groups and give each group a different topic to work on at home.Copy | |

ChatGPT | |

Of course! Here are some topic ideas related to mathematical sequences for students to work on in groups. Each topic is broad enough to allow students to delve into specific aspects of mathematical sequences. Divide the class into 5 groups, each with 4 students, and assign each group one of the following topics. ## Topics to Explore:## Group 1: History of Mathematical Sequences– ## Group 2: Fibonacci Sequence and Its Applications– ## Group 3: Arithmetic and Geometric Sequences– ## Group 4: Monotonicity of Sequences– ## Group 5: Convergence of Sequences– ## Additional Tips:– ## Presentation Schedule:– Assign each group a presentation date so that all groups have enough time to prepare and present their topics over several lessons. I hope these topics will be inspiring and help students deepen their understanding of mathematical sequences. Good luck! | |

You | |

I am a high school math teacher. Please come up with 5 examples to solve from the field of mathematical sequences, monotonic sequences, and recursive sequences. Create 5 easy tasks for beginner students, 5 medium-difficulty tasks, and 5 difficult tasks for advanced students. Copy | |

ChatGPT | |

Of course! Here are examples of tasks related to recursive sequences, divided into three levels of difficulty: easy, medium, and hard. ## Easy Tasks1. 2. 3. 4. 5. ## Medium-Difficulty Tasks1. 2. 3. 4. 5. ## Hard Tasks1. 2. 3. 4. 5. I hope these examples will be useful for your students at different levels of proficiency. Good luck with your teaching! | |

If you have even the smallest information about your students’ interests, you can make good use of it in teaching mathematics.

You | |

I am a high school math teacher. I would like to solve 2 word problems related to mathematical sequences with my class. I know that many boys in my class are interested in football and support our local football team, Pogoń Szczecin. I also have a large group of girls who love horses and are interested in horse breeding. Propose two word problems related to mathematical sequences that consider my students’ interests. Copy | |

ChatGPT | |

Understood, here are more realistic and challenging word problems related to mathematical sequences that consider your students’ interests. ## Problem 1: Football and Pogoń Szczecin
The football club Pogoń Szczecin organizes a contest for its fans. As part of the contest, they give away club gadgets at each match. At the first match, 50 gadgets were given away, and at each subsequent match, the number of gadgets given away increases by 10. a) Find the general formula of the sequence a_n, which describes the number of gadgets given away at the n-th match. b) Calculate how many gadgets will be given away at the 15th match. c) Calculate how many gadgets the club will distribute during an entire season of 30 matches.
a) The general formula for an arithmetic sequence: b) Number of gadgets given away at the 15th match: c) Total number of gadgets given away in 30 matches: ## Problem 2: Kasia and Horse Accessories
Kasia dreams of buying a set of horse accessories that costs 2000 PLN. In the first month, she saved 100 PLN, and each subsequent month she plans to save 50 PLN more than the previous month. a) Find the general formula of the sequence a_n, which describes the amount saved in the n-th month. b) Calculate how much Kasia will save in the sixth month. c) After how many months will Kasia be able to buy the set of horse accessories?
a) The general formula for an arithmetic sequence: b) Amount saved in the sixth month: c) Total savings after n months: To find out when Kasia can buy the set for 2000 PLN, solve the equation: Solving this quadratic equation: Thus, Kasia will be able to buy the set after 8 months, as we round to the nearest whole number. I hope these problems will be interesting and engaging for your students! | |

The king of difficult questions from math students is, of course, “What will this be useful for in my life?” How does ChatGPT handle it in the context of mathematical sequences?

You | |

I am a high school student. How can sequences be useful to me in the future?Copy | |

ChatGPT | |

Mathematical sequences, both arithmetic and geometric, have wide applications in many fields and can be very useful in various careers and life situations. Here are some examples of how sequences can be useful: ## 1. | |

ChatGPT is a polyglot. The quality of its translations is excellent.

You | |

Translate the text below into English: “Wielkie twierdzenie Fermata – twierdzenie teorii liczb, które brzmi: dla liczby naturalnej 𝑛>2 nie istnieją takie liczby naturalne dodatnie 𝑥, 𝑦, 𝑧, które spełniałyby równanie 𝑥^n+𝑦^n=𝑧^𝑛 Równanie to jest znane jako równanie Fermata. Pierre de Fermat zanotował to twierdzenie na marginesie łacińskiego tłumaczenia książki Arithmetica Diofantosa i opatrzył następującą uwagą: znalazłem zaiste zadziwiający dowód tego twierdzenia. Niestety, margines jest zbyt mały, by go pomieścić, lub w innej wersji: Jest niemożliwe rozłożyć sześcian na dwa sześciany, czwartą potęgę na dwie czwarte potęgi i ogólnie potęgę wyższą niż druga na dwie takie potęgi; znalazłem naprawdę zadziwiający dowód tego, jednak margines jest za mały, by go pomieścić.”Copy | |

ChatGPT | |

## Fermat’s Last Theorem – a number theory theorem which states:For any natural number n > 2 , there are no positive natural numbers x , y , and z that satisfy the equation x^n + y^n = z^nThis equation is known as Fermat’s equation. Pierre de Fermat noted this theorem in the margin of his Latin translation of the book *Arithmetica* by Diophantus and added the following remark: “I have discovered a truly marvelous proof of this proposition, which this margin is too narrow to contain.” Or in another version: “It is impossible to separate a cube into two cubes, a fourth power into two fourth powers, and in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which, however, the margin is too narrow to contain.” | |

Finding the transcription of a YouTube video is not difficult. Just turn on subtitles, find and click on “Show transcript” in the description, and then copy the transcript text to the clipboard.

Chat GPT will translate the transcript into your language.

As an example, I used a short lecture video from YouTube.

You | |

Translate the following transcription into Polish: 0:00 (…)Copy | |

ChatGPT | |

0:00 Nowy, duży temat, hura! Więc, 0:02 uh, 0:04 mam nadzieję, że uznacie to za 0:06 trochę oddechu po 0:09 parametrach, które wiele osób uważa za 0:12(…) | |

As with any tool, there are certain rules to follow when using Chat GPT. As a teacher, you also have the duty to set an example for your students on how to use it responsibly.

**Data Privacy:**Never share or save personal data, sensitive information, or content that may violate user privacy. Do not enter your students’ names, surnames, or any other data into Chat GPT.**Safe Use:**Make sure you use Chat GPT in a safe and trusted environment to avoid unauthorized access to the system. Remember that conversations with Chat GPT are archived and do not disappear. If someone uses the same device as you, they may have access to them.**Updates and Security:**Regularly update your software and use appropriate security measures to protect the system from cyber threats. Remember to use antivirus software, secure passwords, and follow all basic security principles.

**Transparency:**Always inform your students that they are talking to AI, not a human.**Impartiality:**Maintain neutrality and do not support or promote specific political, religious, or other ideologies.**Responsibility:**Avoid generating content that may be offensive, harmful, misleading, or unethical.**Compliance with the Law:**Ensure that all actions and generated content comply with applicable laws and regulations.

When writing this article, I used:

- “Basic Matura Course” Anna Zalewska (video course)
- “Limit Course” Krystian Karczyński (video course)
- “The ChatGPT Guide for Business” Mr. Andrew Ward
- “10 Ways to Use ChatGPT in High School and College” Glove Academia
- “The AI Classroom: The Ultimate Guide to Artificial Intelligence in Education (The Everything Edtech Series)” Dan Fitzpatrick, Amanda Fox, Brad Weinstein

Founder and head of the mathematical services MathKiwi and eTrapez.

Graduate of Mathematics at the Poznań University of Technology. Math tutor with many years of experience. Creator of the first eTrapez Courses, which have become extremely popular among students across Poland.

Lives in Szczecin, Poland. Enjoys walks in the forest, beach outings, and kayaking.

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