Phi to the what?

Algebra Level 4

ϕ 15 \phi^{15} can be written as a + b ϕ a+b\phi , where a a and b b are both integers. Find a + b a+b .

Note: ϕ = 1 + 5 2 \phi = \frac{1+\sqrt{5}}{2}


The answer is 987.

Bob Krueger by Bob Krueger , 23, USA

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2 solutions

Shriram Lokhande
Jul 24, 2014

The powers of golden ratio ϕ \phi are written as ϕ n = F n ϕ + F n − 1 \phi^n=F_n\phi+F_{n-1} Where F_n are the fibonnaci numbers F n + F n − 1 = F n + 1 F_n+F_{n-1}=F_{n+1}

ϕ 15 = F 15 ϕ + F 14 \phi^{15}=F_{15}\phi+F_{14}

we get a + b = F 16 = 987 a+b = F_{16} = \boxed{987}

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Naved Husain
Mar 18, 2014

Phi^{n}=F(n-1)+F(n)phi Where F is Fibonacci number Therefore n=15 Phi^15=F14+F15 phi Phi^15=377+610 phi (Fibonacci 14=377 and fibonacci 15=610) a=377 and b = 610 a+b=987

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