The Fibonacci sequence

Question

The Fibonacci sequence

a1=1,a2=1,a3=2,a4=3,a5=5,a6=8…
is defined recursively using the formulas a1=a2=1 and an+2=an+an+1 for all n≥1. Find the greatest common divisor of a484 and a2013.

Answer 1

It has been proven that

gcd(F_m,F_n) = F_gcd(m,n)

So,

gcd(F₄₈₄,F₂₀₁₃) = F_{gcd(484,2013)} = F_{11_{= 89}}

Answer 2

To find the greatest common divisor (GCD) of a484 and a2013, we can use the Euclidean algorithm.

Step 1: Calculate a2013

First, let's calculate the value of a2013 in the Fibonacci sequence using the recursive formula.

a1 = 1
a2 = 1

We can use a loop to calculate the Fibonacci sequence up to the desired term:

for i in range(3, 2014):
a_i = a_i-1 + a_i-2
a2013 = a_i

By iterating through the loop, we can find the value of a2013.

Step 2: Calculate a484

Similarly, let's calculate the value of a484 in the Fibonacci sequence using the same loop:

for i in range(3, 485):
a_i = a_i-1 + a_i-2
a484 = a_i

Step 3: Find the GCD

Now that we have the values of a484 and a2013, we can find their GCD.

We can use the Euclidean algorithm, which states that the GCD of two numbers is equal to the GCD of the smaller number and the difference between the larger number and the smaller number.

GCD(a484, a2013) = GCD(a484, a2013 - a484)

Repeat this process until we reach 0:

GCD(a484, a2013) = GCD(a484, a2013 - a484)
GCD(a484, a2013 - a484) = GCD(a484, (a2013 - a484) % a484)
GCD(a484, (a2013 - a484) % a484) = GCD(a484, (a2013 % a484))

Continue this process until the remainder reaches 0. The final non-zero remainder will be the GCD of a484 and a2013.

This can be implemented in code as follows:

def gcd(a, b):
while b != 0:
temp = b
b = a % b
a = temp
return a

gcd_result = gcd(a484, a2013)

The value of gcd_result will be the GCD of a484 and a2013.