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Arithmetic Sequences Basics
Ever notice how some number patterns follow a simple "add the same amount each time" rule? That's an arithmetic sequence! In sequences like 2, 7, 12, 17, 22, ... each number increases by 5.
An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. This consistent difference is called the common difference (often written as "d").
To check if a sequence is arithmetic, just subtract each term from the one that follows it. If you always get the same answer, it's arithmetic! For example, in -5, -3, -1, 1, 3,...:
- (-3) - (-5) = 2
- (-1) - (-3) = 2
- 1 - (-1) = 2
- 3 - 1 = 2
💡 The key to identifying arithmetic sequences is consistency! If the difference between consecutive terms stays the same throughout the entire sequence, you're looking at an arithmetic sequence.
Finding the nth Term
Once you know a sequence is arithmetic, you can find any term without writing out the whole sequence. That's super helpful when you need the 100th term!
The formula for finding the nth term is: an = a₁ + d
- a₁ is your first term
- d is your common difference
- n is which term you're looking for
For example, with the sequence 57, 45, 33, 21,...:
- First find d: 45 - 57 = -12 (each term decreases by 12)
- Identify a₁ = 57
- Use the formula: aₙ = 57 + (-12)
To find the 19th term, just plug in n = 19: a₁₉ = 57 + (19-1)(-12) = 57 + 18(-12) = 57 - 216 = -159
💡 Notice the pattern: the coefficient of d is always one less than the term number. This makes sense because you start with a₁ and then add d repeatedly.
Solving for Missing Information
You can use the arithmetic sequence formula to solve for any missing piece - whether that's the first term, the common difference, or a specific term number.
When you're given the nth term (aₙ) and the common difference (d), you can find the first term (a₁):
- Start with the formula: aₙ = a₁ + d
- Substitute what you know
- Solve for a₁
For example, if a₁₁ = 41 and d = 5: 41 = a₁ + (11-1)(5) 41 = a₁ + 50 a₁ = -9
When given two different terms, you can set up a system:
- a₆ = 7 means a₁ + 5d = 7
- a₂₂ = 87 means a₁ + 21d = 87
- Solving these equations gives a₁ = -18 and d = 5
💡 Think of these problems as puzzles! You have a formula with multiple variables, and each piece of information helps you narrow down the missing values.
Arithmetic Series
What happens when you want to add up all the terms in an arithmetic sequence? That's an arithmetic series!
The difference between a sequence and a series:
- A sequence is a list of numbers (1, 3, 5, 7, 9,...)
- A series is the sum of those numbers (1 + 3 + 5 + 7 + 9 + ...)
Luckily, there's a formula to find the sum quickly: Sₙ = Where:
- n is the number of terms
- a₁ is the first term
- aₙ is the last term
For example, to find the sum of the first 15 terms where aₙ = 9 + 3n:
- Find a₁ = 9 + 3(1) = 12
- Find a₁₅ = 9 + 3(15) = 54
- Apply the formula: S₁₅ = (15/2)(12 + 54) = (15/2)(66) = 495
💡 This formula saves you tons of time! Instead of adding 15 numbers, you only need to know the first term, last term, and how many terms you're adding.
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 Knowunity AI. 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 Knowunity AI. 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
Understanding Arithmetic Sequences and Series
Arithmetic sequences and series are patterns where numbers increase or decrease by a consistent amount. These patterns show up everywhere in math and real life - from calculating interest to predicting patterns. Understanding how to work with these sequences will... Show more
Arithmetic Sequences Basics
Ever notice how some number patterns follow a simple "add the same amount each time" rule? That's an arithmetic sequence! In sequences like 2, 7, 12, 17, 22, ... each number increases by 5.
An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. This consistent difference is called the common difference (often written as "d").
To check if a sequence is arithmetic, just subtract each term from the one that follows it. If you always get the same answer, it's arithmetic! For example, in -5, -3, -1, 1, 3,...:
- (-3) - (-5) = 2
- (-1) - (-3) = 2
- 1 - (-1) = 2
- 3 - 1 = 2
💡 The key to identifying arithmetic sequences is consistency! If the difference between consecutive terms stays the same throughout the entire sequence, you're looking at an arithmetic sequence.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Finding the nth Term
Once you know a sequence is arithmetic, you can find any term without writing out the whole sequence. That's super helpful when you need the 100th term!
The formula for finding the nth term is: an = a₁ + d
- a₁ is your first term
- d is your common difference
- n is which term you're looking for
For example, with the sequence 57, 45, 33, 21,...:
- First find d: 45 - 57 = -12 (each term decreases by 12)
- Identify a₁ = 57
- Use the formula: aₙ = 57 + (-12)
To find the 19th term, just plug in n = 19: a₁₉ = 57 + (19-1)(-12) = 57 + 18(-12) = 57 - 216 = -159
💡 Notice the pattern: the coefficient of d is always one less than the term number. This makes sense because you start with a₁ and then add d repeatedly.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Solving for Missing Information
You can use the arithmetic sequence formula to solve for any missing piece - whether that's the first term, the common difference, or a specific term number.
When you're given the nth term (aₙ) and the common difference (d), you can find the first term (a₁):
- Start with the formula: aₙ = a₁ + d
- Substitute what you know
- Solve for a₁
For example, if a₁₁ = 41 and d = 5: 41 = a₁ + (11-1)(5) 41 = a₁ + 50 a₁ = -9
When given two different terms, you can set up a system:
- a₆ = 7 means a₁ + 5d = 7
- a₂₂ = 87 means a₁ + 21d = 87
- Solving these equations gives a₁ = -18 and d = 5
💡 Think of these problems as puzzles! You have a formula with multiple variables, and each piece of information helps you narrow down the missing values.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Arithmetic Series
What happens when you want to add up all the terms in an arithmetic sequence? That's an arithmetic series!
The difference between a sequence and a series:
- A sequence is a list of numbers (1, 3, 5, 7, 9,...)
- A series is the sum of those numbers (1 + 3 + 5 + 7 + 9 + ...)
Luckily, there's a formula to find the sum quickly: Sₙ = Where:
- n is the number of terms
- a₁ is the first term
- aₙ is the last term
For example, to find the sum of the first 15 terms where aₙ = 9 + 3n:
- Find a₁ = 9 + 3(1) = 12
- Find a₁₅ = 9 + 3(15) = 54
- Apply the formula: S₁₅ = (15/2)(12 + 54) = (15/2)(66) = 495
💡 This formula saves you tons of time! Instead of adding 15 numbers, you only need to know the first term, last term, and how many terms you're adding.
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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Most popular content in Algebra 1
Square Roots And Cube Roots
The square root of a number means the given number is raised to the power ½, and it is denoted by the symbol '√' called the radical sign.
Algebra 16th
Most popular content
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.6/5
App Store
4.7/5
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 Knowunity AI. 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 Knowunity AI. 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