5 ways to derive the general term of Fibonacci sequence

What if you are told to find the 100th Fibonacci number? Do you start from the first two terms? Wouldn't it be better if you know the general term of Fibonacci sequence, and just plug in n = 100 to obtain the answer? There are 4 different derivations presented in the video, and see which one is your favorite.

I have also rated these methods in terms of satisfaction, and of course, I provided some reasons for the rating, but all these ratings are definitely just my opinions, and you have every right to disagree with me, and maybe tell me why in the comments!

These methods include (1) just directly checking by induction, (2) characteristic equation for recurrence relations (i.e. combining geometric sequences as called in the video), (3) playing with the golden ratio phi, (4) matrix multiplication (shown in a separate video as linked above), and (5) generating function.

All these are valid proofs, but they differ in the level of satisfaction! Mathematics is not just about whether something is right or wrong; it can also be about discovering a better, more satisfying proof.

Sources:

(Penultimate derivation)
(First derivation)
(Ultimate derivation)

Useful links:
(More about generating functions)

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#mathemaniac #math #fibonacci #generalterm #binetformula #generatingfunction #goldenratio

Video Chapters:
00:00 Intro
00:59 Checking by induction
02:12 Combining geometric sequences
04:17 From the golden ratio
05:47 Matrix multiplication
05:58 Generating function
09:29 Google form
10:06 Endcard stuff

[mathemaniac]

Next video: "Untold connection: LaGrange and an ancient Chinese problem"
It will be uploaded on 12th July. There will then be a follow-up video uploaded on 19th July.

If you know linear algebra, be sure to also check out the other video (first link in the description or the end-screen video on the right).

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[tsunningwah]

Love your vids but I have a question regarding the first proof. Well, as you have mentioned, fn is not a geometric sequence. But assuming it is could lead us to the correct general term. So, why assuming a false premise could bring us to a correct conclusion??

[abbasmehdi]

Much informative...keep continue and never stop making such nice vedios

[khansahab]

Nice ....you are a nice MAN bro. Keep up the great content...very less people are like you

[abeginner]

Hello sir
I am from India(a JEE student) and I have a completely different solution for the general term of Fibonacci series.
Reply if you trust me and want to know that solution....

[theimmux]

I derived the Binet formula through partial sums. Let's just say that the proof isn't very compact with all the algebra.