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Dec 18, 2025
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Arithmetic Sequences and Series: Essential Patterns and Their Applications
Knowunity Philippines
@knowunityphilippines
1 / 10
Getting Started with Arithmetic Sequences
Think about counting by 5s: 5, 10, 15, 20... or a weekly allowance that increases by £2 each week. These are arithmetic sequences - number patterns where you add the same amount each time.
An arithmetic sequence is simply a list of numbers where each term after the first is found by adding a constant value called the common difference. The pattern looks like this: first term, first term + difference, first term + 2×difference, and so on.
Here's what makes it easy to spot: if you subtract any term from the next term, you'll always get the same answer. For example, in 3, 7, 11, 15, the difference is always 4 (7-3=4, 11-7=4, 15-11=4).
Quick tip: The common difference can be positive (increasing sequence), negative (decreasing sequence), or even zero (all terms the same)!
Finding Any Term with the Formula
You don't need to write out every single term to find the 50th or 100th term - there's a brilliant shortcut! The nth term formula is your mathematical superpower: aₙ = a₁ + d.
Let's break this down: aₙ is the term you want to find, a₁ is the first term, n is the position, and d is the common difference. It's like having a calculator that jumps straight to any term you need.
Say a worker earns £500 on day one, with £25 increases daily. To find day 10's wage: a₁₀ = 500 + (10-1)×25 = 500 + 225 = £725. No need to calculate days 2 through 9!
You can also work backwards - if you know a later term, you can find the first term or common difference by rearranging the formula.
Remember: This formula works for any arithmetic sequence, making complex calculations surprisingly simple!
Adding It All Up: Arithmetic Series
An arithmetic series is what happens when you add up all the terms in an arithmetic sequence. Instead of tediously adding each term, mathematicians have given us two powerful formulas.
The first formula works when you know the first and last terms: Sₙ = n/2 × . This uses the clever fact that pairs of terms from opposite ends always add up to the same total.
The second formula is perfect when you know the first term and common difference: Sₙ = n/2 × . You don't even need to find the last term first!
For example, if a farmer plants 15 plants in the first row and adds 3 more plants in each subsequent row, the first 12 rows contain: S₁₂ = 12/2 × = 6 × 63 = 378 plants total.
Pro tip: Both formulas give the same answer - choose whichever one matches the information you're given!
Real-World Problem Solving
These aren't just textbook concepts - arithmetic sequences appear everywhere in business, construction, and daily life. The key is recognizing the patterns and applying the right approach.
For salary problems, think about annual increases. A teacher starting at £25,000 with £2,000 yearly increases creates an arithmetic sequence for total yearly earnings. Over 5 years, you'd calculate each year's total salary, then sum them up.
Stadium seating is another classic example. If the first row has 20 seats and each row adds 4 more seats, you can find the total capacity using the series formula instead of counting every single seat.
When tackling these problems, always identify what you're looking for first, list your known information, choose the appropriate formula, substitute your values, and check that your answer makes sense in context.
Success strategy: Real-world problems often hide the sequence pattern in a story - your job is to extract the mathematical relationships!
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Most popular content in GenMath
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Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.9/5
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4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Arithmetic Sequences and Series: Essential Patterns and Their Applications
Knowunity Philippines
@knowunityphilippines
Ever wondered how salary increases, stadium seating, or even jeepney schedules follow predictable patterns? Arithmetic sequences and series are mathematical patterns where numbers increase or decrease by the same amount each time, and they're everywhere in real life.
Getting Started with Arithmetic Sequences
Think about counting by 5s: 5, 10, 15, 20... or a weekly allowance that increases by £2 each week. These are arithmetic sequences - number patterns where you add the same amount each time.
An arithmetic sequence is simply a list of numbers where each term after the first is found by adding a constant value called the common difference. The pattern looks like this: first term, first term + difference, first term + 2×difference, and so on.
Here's what makes it easy to spot: if you subtract any term from the next term, you'll always get the same answer. For example, in 3, 7, 11, 15, the difference is always 4 (7-3=4, 11-7=4, 15-11=4).
Quick tip: The common difference can be positive (increasing sequence), negative (decreasing sequence), or even zero (all terms the same)!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Finding Any Term with the Formula
You don't need to write out every single term to find the 50th or 100th term - there's a brilliant shortcut! The nth term formula is your mathematical superpower: aₙ = a₁ + d.
Let's break this down: aₙ is the term you want to find, a₁ is the first term, n is the position, and d is the common difference. It's like having a calculator that jumps straight to any term you need.
Say a worker earns £500 on day one, with £25 increases daily. To find day 10's wage: a₁₀ = 500 + (10-1)×25 = 500 + 225 = £725. No need to calculate days 2 through 9!
You can also work backwards - if you know a later term, you can find the first term or common difference by rearranging the formula.
Remember: This formula works for any arithmetic sequence, making complex calculations surprisingly simple!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Adding It All Up: Arithmetic Series
An arithmetic series is what happens when you add up all the terms in an arithmetic sequence. Instead of tediously adding each term, mathematicians have given us two powerful formulas.
The first formula works when you know the first and last terms: Sₙ = n/2 × . This uses the clever fact that pairs of terms from opposite ends always add up to the same total.
The second formula is perfect when you know the first term and common difference: Sₙ = n/2 × . You don't even need to find the last term first!
For example, if a farmer plants 15 plants in the first row and adds 3 more plants in each subsequent row, the first 12 rows contain: S₁₂ = 12/2 × = 6 × 63 = 378 plants total.
Pro tip: Both formulas give the same answer - choose whichever one matches the information you're given!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Real-World Problem Solving
These aren't just textbook concepts - arithmetic sequences appear everywhere in business, construction, and daily life. The key is recognizing the patterns and applying the right approach.
For salary problems, think about annual increases. A teacher starting at £25,000 with £2,000 yearly increases creates an arithmetic sequence for total yearly earnings. Over 5 years, you'd calculate each year's total salary, then sum them up.
Stadium seating is another classic example. If the first row has 20 seats and each row adds 4 more seats, you can find the total capacity using the series formula instead of counting every single seat.
When tackling these problems, always identify what you're looking for first, list your known information, choose the appropriate formula, substitute your values, and check that your answer makes sense in context.
Success strategy: Real-world problems often hide the sequence pattern in a story - your job is to extract the mathematical relationships!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user