A Golden Radical and Powers of Golden Ratio

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[resilientcerebrum]

Fibonacci sequence is what we comment as the remainder when that pattern is divided by golden ratio

[antoniopedrofalcaolopesmor]

I'm impressed you read our comments and still give feedback, that's the good thing of having 50 or so comments instead of thousands. One day when you'll grow bigger in subscribers and viewers, which I'm certain it will happen due to the competence and merit of the videos you so kindly share with us, you no longer will have the time to answer the thousands of comments, after all growing comes also linked to some less positive things, it is not only advantages, but this is not a perfect world. Anyway you deserve to have a greater share of viewers and subscribers and that will certainly motivate you to keep up the good work, so keep going and sharing these nice videos with us, thanks a lot!

[robertingliskennedy]

nice pace, good style, both watchable & listenable, thanks

[borisedgartribenouyuni]

I have a special feeling about this number. What a great video man. You're awesome

[aliscais]

I didn't know about this feature of golden ratio. Thank you for video.
When I was solving this equation, I saw that x - a = 1 + a = a^2, so x = 1 + 2a

[MathElite]

Missed the premiere...rip
Nice equation

[aashsyed]

1st! golden prize again! looks cool! AWESOME

[bakixanmadatov]

Hello!
In which video program do you use these examples? What is the name of the video program?

[avneeshabbi]

Talking about the golden ratio, the nth power of the golden ratio can be written as (golden ratio)^n= F(n+1) * (golden ratio) + F(n) where F(n) is the Fibonacci sequence defined as F(1)= 1, F(2)= 1 and F(n+2)= F(n+1) + F(n) where n is a natural number greater than or equal to 1.

[satyapalsingh]

Nice video.Golden ratio is (sqrt5+1)/2=1.618=phi.You are genius, dear professor .

[ramaprasadghosh]

Writing y for LHS one gets
y*y = x- y
or ( y+1/2)^2 = x +1/4
or y = (√(4x+1) -1)/2
y= -(√(4x+1) +1)/2
similarly writing z for RHS one gets
z*z = 1 + z
or (z-1/2)^2 = 5/4
or z = (1+√5)/2
Hereby
(√(4x+1) -1)/2 = (1+√5)/2
or
√(4x+1) = 2+√5
or 4x =( 2+√5)^2 -1
or x = 2 +√5

[luggepytt]

Hint! Here’s a trick that my favorite mathematician (SyberMath) has used a lot in other videos.

We had the equation φ² = φ + 1 earlier (in the form a² = a + 1), so we can substitute φ + 1 for φ² in the formula for x. That spares us some calculations:

x = φ² + φ
x = (φ +1) + φ
x = 2φ + 1

(Edit: Maybe I should have watched the whole video before commenting. I see now that you explore this at the end.)

[zalut_sky]

cool problem, interesting facts about phi also

[skylardeslypere]

Wow, I never knew that the powers of phi were related to the Fibonacci numbers :0
Then again, it's pretty easy to prove, using induction, that φⁿ = F(n) φ + F(n-1) where F(n) represents the n-th fibonacci number.

[manojsurya]

The properties were more amazing than the problem itself, I already seen the 2nd property in Math Elite video, and the 1st one was amazing too

[ThAlEdison]

I'd like to see your take on using generating functions to get the closed form of the Fibbonacci numbers.

[stephenstafford]

Yes the negative solution is wrong—you can plug it in to verify. But the fact it is negative alone is not proof it is wrong.

“Lets think critically” just did a math video explaining that:

[resilientcerebrum]

You have come 1-3rd way to the number of subscribers you will get this year ... I am from future

[peterkiedron]

Strange. Powers of fi is a geometry sequence and at the same time an arithmetic sequence. Does not look right.