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Dec 11, 2025
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Knowunity Philippines
@knowunityphilippines
Geometric sequences are everywhere around you - from population growth... Show more
Think of geometric sequences as number patterns where you multiply (not add) to get from one term to the next. The magic number you multiply by is called the common ratio (r), and it stays the same throughout the entire sequence.
To spot a geometric sequence, check if the ratio between consecutive terms is constant. For example, in the sequence 2, 6, 18, 54, 162..., you'll find r = 6/2 = 3, and 18/6 = 3, and 54/18 = 3. Since the ratio is always 3, this is definitely geometric!
The common ratio determines how your sequence behaves. When r > 1, your numbers grow rapidly (like compound interest). When 0 < r < 1, they shrink (like radioactive decay). When r < 0, your sequence bounces between positive and negative values - quite dramatic!
Real-world connection: If prices in the Philippines increase by 5% yearly, starting at ₱100, you get ₱100, ₱105, ₱110.25, ₱115.76... - a perfect geometric sequence with r = 1.05.
Why calculate every single term when you can jump straight to any position? The geometric sequence formula is your shortcut: aₙ = a₁ × rⁿ⁻¹
Here's your step-by-step approach: identify the first term (a₁), find the common ratio (r), decide which term you need (n), then substitute everything into the formula. For instance, finding the 8th term of 3, 12, 48, 192... gives you a₈ = 3 × 4⁷ = 49,152.
This formula becomes absolutely essential when dealing with large numbers. Imagine manually calculating the 50th term - you'd be there all day! The formula does it in seconds.
Study tip: Practice rearranging this formula to find missing terms or unknown ratios - it's a common exam trick that catches many students off guard.
A geometric series happens when you add up the terms of a geometric sequence, rather than just listing them. This is where things get really practical - think total savings, total distance, or total population over time.
For finite geometric series (adding a specific number of terms), use Sₙ = a₁/. This formula saves you from the nightmare of adding dozens of terms manually.
Consider Maria's savings plan where she doubles her monthly savings: ₱500, ₱1000, ₱2000, ₱4000, ₱8000, ₱16000. Using the formula with a₁ = 500, r = 2, n = 6, her total savings over 6 months equals ₱31,500. Much easier than adding each month individually!
Exam alert: When r = 1 (all terms equal), the formula doesn't work - just multiply the first term by the number of terms instead.
Here's something mind-blowing: some infinite geometric series (that go on forever) actually have finite sums! This only works when the absolute value of your common ratio is less than 1, using S∞ = a₁/.
Picture a ball dropped from 10 metres that bounces to 60% of its previous height each time. The total distance travelled isn't infinite - it's exactly 40 metres! The ball travels down 10m initially, then keeps bouncing up and down in a geometric pattern.
This concept appears in loads of real situations: loan calculations, repeating decimals, and even population models. When |r| ≥ 1, the series diverges (goes to infinity), which makes sense - if something keeps growing or stays the same size, the total will be massive.
Quick check: Always verify |r| < 1 before using the infinite sum formula, or you'll get nonsensical answers in your exams.
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.
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
Knowunity Philippines
@knowunityphilippines
Geometric sequences are everywhere around you - from population growth to your savings account interest, and even in the way a bouncing ball loses height. Unlike arithmetic sequences that add the same number each time, geometric sequences multiply by a... Show more
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Think of geometric sequences as number patterns where you multiply (not add) to get from one term to the next. The magic number you multiply by is called the common ratio (r), and it stays the same throughout the entire sequence.
To spot a geometric sequence, check if the ratio between consecutive terms is constant. For example, in the sequence 2, 6, 18, 54, 162..., you'll find r = 6/2 = 3, and 18/6 = 3, and 54/18 = 3. Since the ratio is always 3, this is definitely geometric!
The common ratio determines how your sequence behaves. When r > 1, your numbers grow rapidly (like compound interest). When 0 < r < 1, they shrink (like radioactive decay). When r < 0, your sequence bounces between positive and negative values - quite dramatic!
Real-world connection: If prices in the Philippines increase by 5% yearly, starting at ₱100, you get ₱100, ₱105, ₱110.25, ₱115.76... - a perfect geometric sequence with r = 1.05.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Why calculate every single term when you can jump straight to any position? The geometric sequence formula is your shortcut: aₙ = a₁ × rⁿ⁻¹
Here's your step-by-step approach: identify the first term (a₁), find the common ratio (r), decide which term you need (n), then substitute everything into the formula. For instance, finding the 8th term of 3, 12, 48, 192... gives you a₈ = 3 × 4⁷ = 49,152.
This formula becomes absolutely essential when dealing with large numbers. Imagine manually calculating the 50th term - you'd be there all day! The formula does it in seconds.
Study tip: Practice rearranging this formula to find missing terms or unknown ratios - it's a common exam trick that catches many students off guard.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
A geometric series happens when you add up the terms of a geometric sequence, rather than just listing them. This is where things get really practical - think total savings, total distance, or total population over time.
For finite geometric series (adding a specific number of terms), use Sₙ = a₁/. This formula saves you from the nightmare of adding dozens of terms manually.
Consider Maria's savings plan where she doubles her monthly savings: ₱500, ₱1000, ₱2000, ₱4000, ₱8000, ₱16000. Using the formula with a₁ = 500, r = 2, n = 6, her total savings over 6 months equals ₱31,500. Much easier than adding each month individually!
Exam alert: When r = 1 (all terms equal), the formula doesn't work - just multiply the first term by the number of terms instead.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Here's something mind-blowing: some infinite geometric series (that go on forever) actually have finite sums! This only works when the absolute value of your common ratio is less than 1, using S∞ = a₁/.
Picture a ball dropped from 10 metres that bounces to 60% of its previous height each time. The total distance travelled isn't infinite - it's exactly 40 metres! The ball travels down 10m initially, then keeps bouncing up and down in a geometric pattern.
This concept appears in loads of real situations: loan calculations, repeating decimals, and even population models. When |r| ≥ 1, the series diverges (goes to infinity), which makes sense - if something keeps growing or stays the same size, the total will be massive.
Quick check: Always verify |r| < 1 before using the infinite sum formula, or you'll get nonsensical answers in your exams.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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.
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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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