The Fibonacci Numbers are also applied in Pascal's Triangle. Entry is sum of the two numbers either side of it, but in the row above. Diagonal sums in Pascal's Triangle are the Fibonacci numbers.

An interesting property of Pascal's Triangle is that its diagonals sum to the Fibonacci sequence, as shown in the picture below:

Fig.2 diagonal sums in Pascal's triangle

The sum of the entries in the nth diagonal of Pascal's triangle is equal to the nth Fibonacci numbers for all positive integers n. Suppose ∑dn sum of the numbers at nth diagonal and fn = nth Fibonacci number, n ≥ 0.

Principle of Mathematical Induction will be used for proof.

For n = 0, ∑d0 = 0, f0 = 0 (The result is true for n = 0) Assume that ∑dk = fk fk+1 = fk+ fk−1, Property of Fibonacci Sequence f(k) = ∑dk and f(k−1) = ∑dk−1, inductive hypothesis

Therefore, ∑dk+1 = ∑dk+∑dk−1

F12+F22 = 12