In this post, I will be outlining the proof for Cassini’s Identity, about which I had discussed in my post on the Jigsaw Paradox. I will be showing 2 methods of proof:
- Proof By Induction
- Proof Using Matrices
Cassini’s Identity
Cassini’s Identity states that for any 3 consecutive Fibonacci numbers
Where Fn is the nth Fibonacci number
Cassini’s identity is a more specific form of Catalan’s Identity which states:
Substituting r=1 in Catalan’s Identity gives Cassini’s Identity.
Proof By Induction
In proof by induction, we show that the statement is valid for the base case i.e n = 1. Then, assuming it is true for n = m, we show that it is true for n = m+1.
The identity is valid for n=1. Now, let us assume its valid for some n = m, i.e.:
We need to show that the identity is valid for n = m+1. In short, we need to prove that:
We know that the recurrence relation holds for Fibonacci series:
Which is what we had to prove.
We have proved Cassini’s Identity using induction. The next proof is relatively simpler and involves matrices.
Proof Using Matrices
The Fibonacci numbers can be represented using matrices in the following manner. If you have any 2 numbers of the Fibonacci numbers, you can calculate the next 2 numbers by just multiplying with the matrix A specified below:
Say I have 55 & 89, 2 consecutive numbers of the Fibonacci series. Multiplying them with the above matrix gives me
You get 89 & 144, the next two numbers in the series. In this way, we can iteratively find the subsequent numbers in the Fibonacci series.
One interesting thing about this matrix is that, it satisfies the following property for n>=1:
There is a very easy proof for this using induction which you can try out on your own.
The determinants of matrices have a very important property, det(XY) = det(X)det(Y). Similarly, det(An) = [det(A)]n
We see that using matrices, the proof was shorter and we could complete it in 2 steps.
There are a few more ways to prove Cassini’s Identity. For instance, one method uses Binet’s formula, which connects the golden ratio with the Fibonacci Series. You can find them in the 2nd reference link.
Thanks for reading!!!
The Other Side Of Mathematics 