Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
GenMath
Dec 21, 2025
1,159
6 pages
Ross Guen Pagarigan Payumo @rossguen
Ever wondered how math appears everywhere in nature, from flower petals to seashells? The Fibonacci sequence and mathematical... Show more
You've probably seen this pattern without even realizing it! The Fibonacci sequence starts with 0, 1, then each number is simply the sum of the two before it 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...
The magic formula is Xₙ = Xₙ₋₁ + Xₙ₋₂, which just means any term equals the sum of the previous two terms. For example, to find the 8th term X₈ = X₇ + X₆ = 13 + 8 = 21.
Here's where it gets mind-blowing when you divide any Fibonacci number by the one before it, you get closer and closer to the golden ratio (φ = 1.618...). Try it yourself - 233 ÷ 144 = 1.6180556, which is incredibly close to φ!
Cool Fact You can arrange Fibonacci numbers into squares that create a perfect spiral - the same spiral found in nautilus shells and galaxies!
Here's a shortcut that'll blow your mind you can calculate any Fibonacci number by multiplying the previous one by the golden ratio and rounding! For numbers above 1, just take 8 × 1.618 = 12.944, which rounds to 13.
The sequence has incredible hidden patterns too. Every 3rd number is divisible by 2, every 4th number by 3, every 5th number by 5 - this pattern continues forever! You can even extend the sequence backwards into negative numbers using the formula X₋ₙ = (-1)ⁿ⁺¹ × Xₙ.
The golden ratio appears everywhere in art, architecture, and nature because our brains find this proportion naturally beautiful and harmonious.
Test Tip Remember that larger Fibonacci pairs give better approximations of the golden ratio - this often appears in exam questions!
Leonardo Fibonacci (1170-1250) wasn't actually the first to discover "his" sequence - it was known in India centuries earlier! His real contribution was introducing Hindu-Arabic numerals (0,1,2,3...) to Europe, replacing clunky Roman numerals. We even celebrate Fibonacci Day on November 23rd (11/23) because of the digits 1,1,2,3.
Now let's tackle arithmetic progressions (A.P.) - sequences where you add the same number each time. Think 2, 5, 8, 11, 14... (adding 3 each time). The pattern is a, a+d, a+2d, a+3d... where 'a' is the first term and 'd' is the common difference.
Two key formulas you absolutely need nth term tₙ = a + d and sum of n terms Sₙ = n/2. These formulas are your best friends for solving A.P. problems quickly!
Memory Trick The sum formula has two versions - use Sₙ = n/2 when you know both ends!
Let's work through A.P. examples that commonly appear on tests. If the first term is 2 and common difference is 3, then tₙ = 2 + 3 = 3n-1. For the 15th term of 7, 17, 27..., you get t₁₅ = 7 + (14×10) = 147.
Geometric progressions (G.P.) work differently - you multiply by the same number each time. In 1, 2, 4, 8, 16..., you multiply by 2. The pattern is a, ar, ar², ar³... where 'r' is the common ratio.
Essential G.P. formulas nth term tₙ = arⁿ⁻¹ and sum Sₙ = a/ when r>1, or Sₙ = a/ when r<1. The key is identifying whether r is greater or less than 1!
Exam Strategy Always find the common ratio first by dividing any term by the previous term - this determines which formula to use!
Finding the common ratio is your first step in any G.P. problem. For 16, 24, 36, 54..., divide 24÷16 = 1.5, so r = 1.5. Remember, the ratio should be the same between any consecutive terms!
Let's solve a tricky one find the 10th term of 2/5, 4/25, 8/125... Here a₁ = 2/5 and r = 2/5, so t₁₀ = (2/5)(2/5)⁹ = 2¹⁰/5¹⁰. Don't let fractions intimidate you - the same rules apply!
For verification, always check your work by calculating a few terms manually. If a₁ = 4 and r = 3, then t₂ = 12, t₃ = 36, t₄ = 108, t₅ = 324, giving S₅ = 484.
Pro Tip When dealing with fractions in G.P., convert everything to the same denominator format to avoid calculation errors!
Here's something amazing some geometric progressions with |r| < 1 actually have a finite sum even with infinite terms! For sequences like 1, 1/2, 1/4, 1/8..., the terms get smaller and smaller, approaching zero.
The sum to infinity formula is beautifully simple S∞ = a/ provided -1 < r < 1. For the sequence 1, 1/2, 1/4, 1/8..., you get S∞ = 1/(1-1/2) = 2. This means adding infinitely many terms gives you exactly 2!
Try this with 1/2 + 1/4 + 1/8 + 1/16 + ... The first term is 1/2, r = 1/2, so S∞ = (1/2)/(1-1/2) = 1. Think about it - you're adding smaller and smaller pieces that together make exactly 1 whole!
Real-World Connection This concept explains how bank interest compounds and why some investments have limited growth potential despite continuous returns!
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.
10
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
Ross Guen Pagarigan Payumo
@rossguen
Ever wondered how math appears everywhere in nature, from flower petals to seashells? The Fibonacci sequence and mathematical progressions reveal the hidden patterns that govern our world, and understanding them will give you powerful tools for solving complex problems.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
You've probably seen this pattern without even realizing it! The Fibonacci sequence starts with 0, 1, then each number is simply the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...
The magic formula is Xₙ = Xₙ₋₁ + Xₙ₋₂, which just means any term equals the sum of the previous two terms. For example, to find the 8th term: X₈ = X₇ + X₆ = 13 + 8 = 21.
Here's where it gets mind-blowing: when you divide any Fibonacci number by the one before it, you get closer and closer to the golden ratio (φ = 1.618...). Try it yourself - 233 ÷ 144 = 1.6180556, which is incredibly close to φ!
Cool Fact: You can arrange Fibonacci numbers into squares that create a perfect spiral - the same spiral found in nautilus shells and galaxies!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Here's a shortcut that'll blow your mind: you can calculate any Fibonacci number by multiplying the previous one by the golden ratio and rounding! For numbers above 1, just take 8 × 1.618 = 12.944, which rounds to 13.
The sequence has incredible hidden patterns too. Every 3rd number is divisible by 2, every 4th number by 3, every 5th number by 5 - this pattern continues forever! You can even extend the sequence backwards into negative numbers using the formula X₋ₙ = (-1)ⁿ⁺¹ × Xₙ.
The golden ratio appears everywhere in art, architecture, and nature because our brains find this proportion naturally beautiful and harmonious.
Test Tip: Remember that larger Fibonacci pairs give better approximations of the golden ratio - this often appears in exam questions!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Leonardo Fibonacci (1170-1250) wasn't actually the first to discover "his" sequence - it was known in India centuries earlier! His real contribution was introducing Hindu-Arabic numerals (0,1,2,3...) to Europe, replacing clunky Roman numerals. We even celebrate Fibonacci Day on November 23rd (11/23) because of the digits 1,1,2,3.
Now let's tackle arithmetic progressions (A.P.) - sequences where you add the same number each time. Think 2, 5, 8, 11, 14... (adding 3 each time). The pattern is a, a+d, a+2d, a+3d... where 'a' is the first term and 'd' is the common difference.
Two key formulas you absolutely need: nth term: tₙ = a + d and sum of n terms: Sₙ = n/2. These formulas are your best friends for solving A.P. problems quickly!
Memory Trick: The sum formula has two versions - use Sₙ = n/2 when you know both ends!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Let's work through A.P. examples that commonly appear on tests. If the first term is 2 and common difference is 3, then tₙ = 2 + 3 = 3n-1. For the 15th term of 7, 17, 27..., you get t₁₅ = 7 + (14×10) = 147.
Geometric progressions (G.P.) work differently - you multiply by the same number each time. In 1, 2, 4, 8, 16..., you multiply by 2. The pattern is a, ar, ar², ar³... where 'r' is the common ratio.
Essential G.P. formulas: nth term: tₙ = arⁿ⁻¹ and sum: Sₙ = a/ when r>1, or Sₙ = a/ when r<1. The key is identifying whether r is greater or less than 1!
Exam Strategy: Always find the common ratio first by dividing any term by the previous term - this determines which formula to use!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Finding the common ratio is your first step in any G.P. problem. For 16, 24, 36, 54..., divide 24÷16 = 1.5, so r = 1.5. Remember, the ratio should be the same between any consecutive terms!
Let's solve a tricky one: find the 10th term of 2/5, 4/25, 8/125... Here a₁ = 2/5 and r = 2/5, so t₁₀ = (2/5)(2/5)⁹ = 2¹⁰/5¹⁰. Don't let fractions intimidate you - the same rules apply!
For verification, always check your work by calculating a few terms manually. If a₁ = 4 and r = 3, then t₂ = 12, t₃ = 36, t₄ = 108, t₅ = 324, giving S₅ = 484.
Pro Tip: When dealing with fractions in G.P., convert everything to the same denominator format to avoid calculation errors!
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 amazing: some geometric progressions with |r| < 1 actually have a finite sum even with infinite terms! For sequences like 1, 1/2, 1/4, 1/8..., the terms get smaller and smaller, approaching zero.
The sum to infinity formula is beautifully simple: S∞ = a/ provided -1 < r < 1. For the sequence 1, 1/2, 1/4, 1/8..., you get S∞ = 1/(1-1/2) = 2. This means adding infinitely many terms gives you exactly 2!
Try this with 1/2 + 1/4 + 1/8 + 1/16 + ... The first term is 1/2, r = 1/2, so S∞ = (1/2)/(1-1/2) = 1. Think about it - you're adding smaller and smaller pieces that together make exactly 1 whole!
Real-World Connection: This concept explains how bank interest compounds and why some investments have limited growth potential despite continuous returns!
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.
10
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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