Using the matrix formula for Fibonacci numbers and some simple calculations, a surprising formula falls out.

The Fibonacci Numbers consist of the sequence of integers: F₀ = 0, F₁ = 1, F₂ = 1, F₃ = 2, F₄ = 3, F₅ = 5, F₆ = 8, F₇ = 13, F₈ = 21, F₉ = 34, F₁₀ = 55, ... . These are defined recursively by the following formula:

What I'm calling the "matrix formula for Fibonacci Numbers" is the following, that we proved by induction:

The next sequence of formulas comes from squaring the above.

Now just take the first equal to the last above:

On the other hand, we can use straight matrix multiplication to get the following:

In the two formulas above, the left hand sides are the same, so the right hand sides are also equal. Equating entries of these two matrices, we get the following. (The right term in the middle equation below comes from substituting F_n+1 = F_n + F_n-1.)

We often just write a top portion of these three equations, since the third equation follows immediately from the first.

Notice that this method has allowed us to discover the above formula, but it was not a proof.