A Nice Algebra Problem | Math Olympiad | Radical Problem

[@quinty.support]

[2/(1+√5)]^12
Consider rationalizing the denominator....
=> [2(1-√5)/(1+√5)(1-√5)]^12
=> [2(1-√5)/(1² - √5²)]^12
=> [2(1-√5)/-4)]^12
=> [(- 1 - √5)/2]^12 => [(1 - √5)/2]^12 (this value is the conjugate of the golden ratio)
Φ = [(1 + √5)/2], /Φ (conjugate of golden ratio) = [(1 - √5)/2]
Φ • /Φ = -1
[/Φ]^12 = [(1 - √5)/2]^12
Let, x = [(1 - √5)/2]......
Φ² = Φ + 1
(since x is the conjugate, it satisfies:)
x² = x + 1
x³ = x • x² = x(x + 1) = x² + x = 2x + 1
x⁴ = x • x³ = x(2x + 1) = 2x³ + x = 3x + 2
x⁵ = x • x⁴ = x(3x + 2) = 3x² + 2x = 5x + 3
x^n = F(n)x + F(n-1)
(°•°) x^12 = F(12)x + F(11)
F(12) = 144 and F(11) = 89
x^12 = 144x + 89
=> x^12 = 144[(1 - √5)/2] + 89
=> x^12 = [(144 - 144√5)/2] + 89
=> x^12 = 72 - 72√5 + 89 = 161 - 72√5

[@SALogics]

Very nice! I really appreciate that ❤

[@musicsubicandcebu1774]

Tricky one

[@SALogics]

Thanks for liking! ❤

[@YAWTon]

Shouldn't you mention the golden ratio and Fibonacci?

[@SALogics]

Yes, ❤

[@trojanleo123]

Answer = 161 - 72√5

[@SALogics]

Very nice! ❤