Antibonacci Sequence #1

Number Theory Level 2

Most math enthusiasts/professionals know the famous Fibonacci Sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... F n = F n − 1 + F n − 2 F_n=F_{n-1}+F_{n-2} F 0 = 0 ; F 1 = 1 F_0=0;~F_1=1

However, not many people have considered going backwards! You, the reader of this problem, are going to become one of those people. Find the value of F − 4 F_{-4} .


The answer is -3.

Blan Morrison by Blan Morrison , 18, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Blan Morrison
Jan 23, 2018

We can work backwards to get F − 4 F_{-4} . starting with calculating F − 1 F_{-1} :

What can I add to 0 to get 1? The answer would be 1 \boxed{1} ! We can rethink our question, by turning it into simple subtraction: 1 − 0 = ? 1-0=? 1 \boxed{1}

From this point, we can generalize:

F n + 2 − F n + 1 = F n F_{n+2}-F_{n+1}=F_n

Keep in mind that the equation above is just a re-written version of the Fibonacci rule: F n = F n − 1 + F n − 2 F_n=F_{n-1}+F_{n-2}

With this generalization, we can repeat the process:

F − 2 = 0 − 1 = − 1 F_{-2}=0-1=-1 F − 3 = 1 − ( − 1 ) = 2 F_{-3}=1-(-1)=2 F − 4 = − 1 − 2 = − 3 F_{-4}=-1-2=-3

Therefore, F − 4 = − 3 \boxed{F_{-4}=-3}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...