Mathematics - Fibonacci Sequence and the Golden Ratio 

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BETTER THAN MY ONLINE TEACHER EXPLAINING FOR 2 HOURS WHILE I GOT IT FOR 24 MINS.

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YOU SAVED ME I just started college and in my senior high school, I didn't come across much math except statistics which I was bad at too. Now with the pandemic, I'm left on my own to learn by myself because my teacher isn't teaching at all and just gives vague videos to watch. Thank you so much. I hope I can come across statistics in your channel.

For those of you confused about at timestamp 16:12 here is the explanation: As the author of the video mentioned, the Fibonacci sequence, apart for the first few numbers, has the property that Fn/Fn-1 = k , where k is the golden ratio. Turns out then that the Fibonacci sequence is a geometric progression, and a property of these progressions is that k^n is the n-th number of the sequence. Now, k in our case (fibonacci sequence) is the golden ration, so k^n is the n-th number of the fibonacci sequence. What we are doing at 16:12 is imposing that our generic progression has the property of fibonacci (so that Fn = Fn-1 + Fn-2), and we are looking to find k (the golden ratio). We actually find two values, because there are two values that satifies the equation F^n = F^(n-1) + F^(n-2). Now the problem is that neither 1.618 nor -0.618 approximates the fibonacci sequence. In fact, for example, 1.618^10 is 123 and not 55. Seems like our calculations are wrong, but not really. We didnt take into account that the first few numbers dont satisfie Fn/Fn-1 = k, and this creates us a problem. How to find the formula for the fibonacci sequence then? Remember that a linear combination of functions that satisfie the fibonacci sequence saties the fibonacci sequence itself too. Also, the possible sequence that can satisfie Fn = Fn-1 + Fn-2 MUST be a combination of the functions 1.618^n and -0.618^n. This because 1.618 and -0.618 are the roots of F^n = F^n-1 + F^n-2. So what we are looking for is a way to combine these 2 functions so that we can actually find the fibonacci sequence. Our fibonacci sequence wants the first term to be 1 and the second term to be 1. (In fact 0 (zero-th term), 1 (first ),1(second)). That's enough to find how to combine our two functions, we just need to solve the following system of equations: c1 * (1.618)^1 + c2 * (-0.618)^1 = 1 (

I watched for almost 10times now in God's grace I understood it... Thank you for this video....

It's odly satisfying, hearing the taps from the calculator ugghhh✨

I just want to thank you for all these videos. Ever since the pandemic started, math hasn't been a fun ride but the way you just explain these things directly just makes it all seem so easy. Wishing you success in you future endeavors, whatever they may be, thanks for helping us get through life in the mathematical aspect!😊

I cannot express how helpful this video was for me! I love how you break everything down to its simplest form. Keep up the great work!!

huge thanks man!!! Im in a horse race rn and my friend is teaching me about the "golden ratio" And i dont understand him but now i know it now ゴゴゴゴ Lesson six

when school is a youtube channel

This channel has taught me more math than Chemistry.

I just straight up love you, best teacher in my life.

MR. Organic Chemistry Tutor, thank you for a fantastic explanation of the Fibonacci Sequence and a powerful derivation of the old Golden Ratio.

Thank you for this tutorial! God bless and more contents to come!

It's nice to see an updated video on this I enjoyed watching it. I'm looking forward to seeing more videos.

Only video where I was actually thankful for the adverts to take a breather 😆

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Ohhh my goodness This helps so much !!! Worth 24mins !!

It helped me understand Fibonacci sequence. Thank you😊

Thank for this 😊 I understand clearly

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Best RU-vid channel ever! Thank you so much:)

Hello! We have our Fibonacci distribution in our book, and we also have an assignment about it. My teacher did not discuss anything about it (IDK, is this how college supposed to look like?). The explanations I found in the book are obscure. Thank you because you've explained it and broke the two golden ratios into sections to make it more understandable to students like me.

Thankyou so much .....helps me a lots to answer my module about Fibonacci sequence 😘

Love this! You deserve more attention and appreciation!

If I start at 5, does that mean I add 3 as it must be of sequence or is prior number to add 4?

Very simple and clear.... Thank you so much 🏅🏅🏅🏅

Were having a whole unit on deltas, long multiplication, and this, but i just learned it in less than 1 day

Excellent . The beauty of mathematic. Only with very patient teacher

Thanks! It makes sense finally.

Kinda long video but covers everything THANKS!!

That was perfect. Thank you for sharing your knowledge.God Bless.

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Thank you so much. I have to do a research paper for math and it’s due tomorrow and as I was about to start it I see the notification of ur vid and I was like thank the lord.

You rock it, amazing!! Thank you

You are great! Thank you for this! Very clear, got it!

Wow great video. So easy to learn!

I loved the way you explained it, being an 11 year old i could actually understand 👍

this help me so much thank you!

Said goodbye to higher than average difficulty level of maths almost 5 years ago.. 🤣🤣 a little rusty but understood the calculations. Uncle Fibonacci is lobh ❤️❤️

Thank you, now I understand it!

Thank you for explanation❤️

thank you that was really helpful 🙂😀

Thankyou for helping me in my activity

Sir this video is so useful I'm from malaysia I'm an dlp student just because I'm in dlp class not all math teacher helps us in English but ur explanation is great I understand things so easily

is it possible for the fibonacci sequence to have negative numbers as the given first two terms? Please explain. Thank you!

I wish that you show the complete solution for each. It would be a big help for me to understand the whole thing you discussed.

my ghad your a good teacher I understand it very well

To complete this, you should also show the geometry/trigonometry method of getting the golden ratio.

Great Explanation

Thx for the video used it for my math Olympiad

Thank you this was really helpful.❤️

Love you so much... Thanks!

Hello! I'm curious what is the basis of the formula in getting the exact value of Fn? Thank you!

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Thanks for this, it really helped me a lot. God bless

Greetings everyone, Please does anyone know any textbook that has practice problems on Fibonacci sequence, mathematical Induction and recursive?

Didnt he have another video about the fibonachi numbers and how they interacted with each other? i think i rember that there was something with them being squared in that video.

what type and brand of calculator are you using?

Amazing! So Helpful!

THis is great but did you know using 0,1 as the start is the trivial case and you can actually pick ANY integers you want to start with and get the same result (golden ratio convergence). Can you dig into the deeper meaning of this?

I have using this sequence in my mlm company for team building and team managing , I think it's happening 1st time in the world mlm ( direct selling) industry.

A sequence that is not understood by many and you make this so simple #man of the century

THANKYOU SO MUCHH!

Interesting. Saw the Fibonacci sequence in The Da Vinci Code. Super weird but cool.

Me using this as my advance “reading” material for tomorrow’s lesson in Math ✨

This is very helpful. Thank you!

16:18 I didn't understand how you got this part. From where did this come ? Of course this doesn't work for any number, so If I assume F means fibonacci number then again in the next step you treat it as a normal number raised to a power. Someone who understood this part, please explain.

Thanks for sharing 🙏

Thank you!

Wow this is amazing.

Do we need to divide 8/5 ?

Thanks for teaching me.

Hey I hope this comment gets answered, so I'm trying to find the term F100, and when I put the formula in my calcu and the answer was in a decimal form, is this the final answer or do I literally need to solve it myself without an calculator?

Thank you sa wakas na ka anwer na jud ko

How i wish instructor explained it like you did

Thank youu, my professor ask us to do a advance reading abt this but hahah since im more of a visual learner when it comes to math, i search for it heheh

henlo sa mga hindi naintindihan ang kanilang teacher

Thank you naintindihan ko ng mabuti

sir, how could I find the terms if the sequence starts with any number expect 0 and 1?

How do you get the square root it's hard to understand without showing the solution

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I love so much math but i gon cause of our mathematics teacher who dont really explain step by step Btw. Im so very thankful i watch your video i have interest in math again

Math is my fav subject but after 6 months of quarantine I don't know what happened lol

thank you so much!

Thank you! Thanks to you I solved task of finding first two number of Fibonacci sequence, where 19th (or 18th, idk, task says it is 19th) number equals 45614. Just by dividing 45614 by golden ratio, which you provided the formula for in your video. By the way, answer is 14, 9, ....

If you need a number less than 12 which formula you use?

What was the name of the formula on 9:41 ?

what's the video that came before? or is this video apart of a larger series/playlist?

how did you arrive at the Fn formula?

Thank you a loooot❤

It's better than my teacher explanation on my module I got here minutes only..

If math was taught to me like this I just might have gottten it. They are teaching kids all wrong. Math is spiritual! Thank you! I’m sharing this video btw

thanks for this

Thank you and God bless.

also, what is the significance of the Fibonacci in organic chemistry?

Why would I use the golden ratio to get the next number when i could just add the 2 following numbers.

Thank you thank you thank you!!!!!

1 minute and 48 seconds in and I already understand !?