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7
0
Olamide
04/12/2025
Maths
Year 10 Module 6 - Sequences
208
•
4 Dec 2025
•
Olamide
@olamide_26
Ever wondered how mathematicians predict patterns in numbers? Sequences are... Show more
Think of a sequence as a number pattern with a secret rule - once you crack it, you can predict what comes next! A linear sequence is the simplest type, where you add (or subtract) the same amount each time. For example, 2, 5, 8, 11... follows the pattern of adding 3.
Each number in your sequence is called a term. So in 2, 5, 8, 11..., the number 8 is the third term. The term-to-term rule tells you how to get from one number to the next - it's like having step-by-step directions.
But here's where it gets powerful: the nth term formula lets you jump straight to any position without calculating all the steps in between. For the sequence 2, 5, 8, 11..., the nth term is 3n - 1. Want the 100th term? Just plug in: 3 × 100 - 1 = 299. Dead simple!
Finding the nth term follows three easy steps: find the difference between terms, multiply by n, then work out what to add or subtract to match your first term. Master this, and you'll never be stumped by linear sequences again.
Quick Tip: The difference between consecutive terms in a linear sequence is always the coefficient of n in your formula!
Beyond basic patterns, Fibonacci-type sequences add the two previous terms together (like 1, 1, 2, 3, 5, 8...), whilst geometric sequences multiply by the same number each time, creating rapid growth or shrinkage.
Quadratic sequences are trickier beasts - they don't have a constant difference like linear ones. Instead, you need to look at the second difference (the difference between the differences). When this second difference is constant, you're dealing with a quadratic pattern that involves n².
Working out the nth term of a quadratic sequence requires patience but follows a clear method. Find those second differences, halve that constant value, and multiply by n². Then subtract this pattern from your original sequence to find what's left over. It sounds complex, but with practice, it becomes routine.
Triangular numbers create a beautiful sequence: 1, 3, 6, 10, 15, 21... These come from arranging dots in triangular patterns, where each new triangle adds another row. They appear frequently in exams, so recognising this pattern instantly gives you a real advantage.
The key to mastering quadratic sequences is methodical working. Always show your second differences clearly, substitute values back into your formula to check it works, and remember that every quadratic nth term will contain an n² component.
Exam Hack: If the differences aren't constant but the second differences are, you've got a quadratic sequence - start looking for that n² term!
These advanced sequences might seem daunting initially, but they follow predictable rules. Once you've practised the methods, you'll spot patterns that would have seemed impossible to crack before.
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Quotes from every main character
English LiteratureApp Store
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Olamide
@olamide_26
Ever wondered how mathematicians predict patterns in numbers? Sequences are everywhere - from the spiral of a nautilus shell to your savings account growth. This module breaks down the different types of number patterns you'll encounter and gives you the... Show more
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Think of a sequence as a number pattern with a secret rule - once you crack it, you can predict what comes next! A linear sequence is the simplest type, where you add (or subtract) the same amount each time. For example, 2, 5, 8, 11... follows the pattern of adding 3.
Each number in your sequence is called a term. So in 2, 5, 8, 11..., the number 8 is the third term. The term-to-term rule tells you how to get from one number to the next - it's like having step-by-step directions.
But here's where it gets powerful: the nth term formula lets you jump straight to any position without calculating all the steps in between. For the sequence 2, 5, 8, 11..., the nth term is 3n - 1. Want the 100th term? Just plug in: 3 × 100 - 1 = 299. Dead simple!
Finding the nth term follows three easy steps: find the difference between terms, multiply by n, then work out what to add or subtract to match your first term. Master this, and you'll never be stumped by linear sequences again.
Quick Tip: The difference between consecutive terms in a linear sequence is always the coefficient of n in your formula!
Beyond basic patterns, Fibonacci-type sequences add the two previous terms together (like 1, 1, 2, 3, 5, 8...), whilst geometric sequences multiply by the same number each time, creating rapid growth or shrinkage.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Quadratic sequences are trickier beasts - they don't have a constant difference like linear ones. Instead, you need to look at the second difference (the difference between the differences). When this second difference is constant, you're dealing with a quadratic pattern that involves n².
Working out the nth term of a quadratic sequence requires patience but follows a clear method. Find those second differences, halve that constant value, and multiply by n². Then subtract this pattern from your original sequence to find what's left over. It sounds complex, but with practice, it becomes routine.
Triangular numbers create a beautiful sequence: 1, 3, 6, 10, 15, 21... These come from arranging dots in triangular patterns, where each new triangle adds another row. They appear frequently in exams, so recognising this pattern instantly gives you a real advantage.
The key to mastering quadratic sequences is methodical working. Always show your second differences clearly, substitute values back into your formula to check it works, and remember that every quadratic nth term will contain an n² component.
Exam Hack: If the differences aren't constant but the second differences are, you've got a quadratic sequence - start looking for that n² term!
These advanced sequences might seem daunting initially, but they follow predictable rules. Once you've practised the methods, you'll spot patterns that would have seemed impossible to crack before.
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
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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Explore the fundamentals of mathematical sequences, including arithmetic, geometric, Fibonacci, square, and triangular sequences. Learn how to derive the nth term for both linear and quadratic sequences with clear examples and formulas. This summary is essential for mastering sequence concepts in mathematics.
MathsExplore the fundamentals of sequences and series, including arithmetic and geometric sequences, their properties, and formulas. This summary covers increasing and decreasing sequences, periodic sequences, and provides examples for better understanding. Ideal for students preparing for exams or seeking to strengthen their grasp of mathematical concepts.
MathsExplore the concept of error intervals, including how to determine maximum and minimum values for rounded and truncated numbers. This summary provides clear examples and step-by-step methods for calculating error bounds, essential for mastering numerical accuracy in mathematics.
MathsExplore the methods to determine the nth term of mathematical sequences, including polynomials and quadratics. This summary covers key concepts such as first and second differences, explicit formulas, and practical examples to enhance your understanding. Ideal for students studying mathematical sequences.
MathsHow to find nth term | examples & tutorial
MathsExplore key concepts of recurrence relations through practical word problems, including drug dosage calculations and loan repayments. This summary covers recursive formulas, compound interest, and exponential decay, providing essential insights for higher mathematics students.
MathsQuotes from every main character
English LiteratureApp Store
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Sociology