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Math
Dec 21, 2025
7,787
5 pages
kish andres @kishandres
Ready to master patterns, sequences, and polynomials? This comprehensive guide covers everything from recognizing number patterns to solving... Show more
Ever notice how your allowance might increase by the same amount each week, or how patterns appear everywhere in math? A sequence is simply a set of numbers that follow a specific pattern, and each number in the sequence is called a term.
The most important type you'll encounter is an arithmetic sequence, where you add the same number (called the common difference) to get from one term to the next. For example, in the sequence 2, 5, 8, 11..., you're adding 3 each time, so d = 3.
To find any term in an arithmetic sequence, use this powerful formula an = a₁ + d. Here, a₁ is your first term, n is the position you want, and d is the common difference. Once you memorize this formula, you can jump to any term without writing out the whole sequence!
Quick Tip Always find the common difference first by subtracting any term from the next term .
Sometimes you need to fill in missing terms between two numbers to create an arithmetic sequence - these are called arithmetic means. It's like finding the perfect stepping stones between two points!
An arithmetic series is just the sum of all terms in an arithmetic sequence. The formula Sₙ = n/2 lets you add up tons of terms instantly without doing it one by one. This formula was actually discovered by Karl Gauss when he was just a kid!
Now here's where things get exciting geometric sequences work differently. Instead of adding the same number each time, you multiply by the same number (the common ratio, r). Think 2, 4, 8, 16... where you multiply by 2 each time.
Memory Hack For geometric sequences, use aₙ = a₁(r)ⁿ⁻¹ and find r by dividing any term by the previous term.
Geometric series follow the same idea as arithmetic series, but for geometric sequences. Use Sₙ = a₁/ to find the sum of the first n terms.
Here's something mind-blowing if your common ratio r is between -1 and 1 , you can find the sum of an infinite geometric series using S∞ = a₁/. This means some infinite sequences actually add up to a finite number!
Harmonic sequences are created by taking reciprocals of arithmetic sequences. So if you have 1, 3, 5, 7..., the harmonic sequence would be 1/1, 1/3, 1/5, 1/7...
The Fibonacci sequence is probably the most famous sequence in math 1, 1, 2, 3, 5, 8, 13... Each term equals the sum of the two terms before it. You'll find Fibonacci numbers everywhere in nature!
Fun Fact The Fibonacci sequence appears in flower petals, shell spirals, and even your family tree!
A polynomial is an algebraic expression with variables raised to whole number powers, like 3x² + 5x - 2. The degree is the highest exponent, and the leading coefficient is the number in front of that highest power term.
Always write polynomials in standard form with the highest degree term first. This makes everything cleaner and easier to work with.
Synthetic division is a super fast way to divide polynomials, but it only works when dividing by something like . Just remember three rules the divisor must be in x - c form, write the dividend in standard form, and don't skip any powers (use 0 for missing terms).
The process looks complicated at first, but it's just a pattern of multiply, add, bring down. Practice a few examples and you'll see how much faster it is than long division!
Pro Tip Synthetic division saves tons of time on tests - master this technique and you'll have extra time for other problems.
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
224
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App 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
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
kish andres
@kishandres
Ready to master patterns, sequences, and polynomials? This comprehensive guide covers everything from recognizing number patterns to solving complex polynomial equations. You'll learn the essential formulas and techniques that show up on almost every math test!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Ever notice how your allowance might increase by the same amount each week, or how patterns appear everywhere in math? A sequence is simply a set of numbers that follow a specific pattern, and each number in the sequence is called a term.
The most important type you'll encounter is an arithmetic sequence, where you add the same number (called the common difference) to get from one term to the next. For example, in the sequence 2, 5, 8, 11..., you're adding 3 each time, so d = 3.
To find any term in an arithmetic sequence, use this powerful formula: an = a₁ + d. Here, a₁ is your first term, n is the position you want, and d is the common difference. Once you memorize this formula, you can jump to any term without writing out the whole sequence!
Quick Tip: Always find the common difference first by subtracting any term from the next term .
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sometimes you need to fill in missing terms between two numbers to create an arithmetic sequence - these are called arithmetic means. It's like finding the perfect stepping stones between two points!
An arithmetic series is just the sum of all terms in an arithmetic sequence. The formula Sₙ = n/2 lets you add up tons of terms instantly without doing it one by one. This formula was actually discovered by Karl Gauss when he was just a kid!
Now here's where things get exciting: geometric sequences work differently. Instead of adding the same number each time, you multiply by the same number (the common ratio, r). Think 2, 4, 8, 16... where you multiply by 2 each time.
Memory Hack: For geometric sequences, use aₙ = a₁(r)ⁿ⁻¹ and find r by dividing any term by the previous term.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Geometric series follow the same idea as arithmetic series, but for geometric sequences. Use Sₙ = a₁/ to find the sum of the first n terms.
Here's something mind-blowing: if your common ratio r is between -1 and 1 , you can find the sum of an infinite geometric series using S∞ = a₁/. This means some infinite sequences actually add up to a finite number!
Harmonic sequences are created by taking reciprocals of arithmetic sequences. So if you have 1, 3, 5, 7..., the harmonic sequence would be 1/1, 1/3, 1/5, 1/7...
The Fibonacci sequence is probably the most famous sequence in math: 1, 1, 2, 3, 5, 8, 13... Each term equals the sum of the two terms before it. You'll find Fibonacci numbers everywhere in nature!
Fun Fact: The Fibonacci sequence appears in flower petals, shell spirals, and even your family tree!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
A polynomial is an algebraic expression with variables raised to whole number powers, like 3x² + 5x - 2. The degree is the highest exponent, and the leading coefficient is the number in front of that highest power term.
Always write polynomials in standard form with the highest degree term first. This makes everything cleaner and easier to work with.
Synthetic division is a super fast way to divide polynomials, but it only works when dividing by something like . Just remember three rules: the divisor must be in x - c form, write the dividend in standard form, and don't skip any powers (use 0 for missing terms).
The process looks complicated at first, but it's just a pattern of multiply, add, bring down. Practice a few examples and you'll see how much faster it is than long division!
Pro Tip: Synthetic division saves tons of time on tests - master this technique and you'll have extra time for other problems.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
224
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App 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
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
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