Fibonacci Partial Sums Tricks

TL;DR

This paper explores divisibility properties of Fibonacci-like sequences to develop quick calculation tricks for partial sums, including generalizations to Pell-like sequences and limitations for Jacobstal-like sequences.

Contribution

It uncovers the divisibility properties of partial sums of Fibonacci-like sequences and extends the magic trick concept to other second-order recurrences.

Findings

Maximum Fibonacci divisor of partial sums identified

Magic trick generalized to Pell-like sequences

No similar trick exists for Jacobstal-like sequences

Abstract

The following magic trick is at the center of this paper. While the audience writes the first ten terms of a Fibonacci-like sequence (the sequence following the same recursion as the Fibonacci sequence), the magician calculates the sum of these ten terms very fast by multiplying the 7th term by 11. This trick is based on the divisibility properties of partial sums of Fibonacci-like sequences. We find the maximum Fibonacci number that divides the sum of the Fibonacci numbers 1 through $n$. We discuss the generalization of the trick for other second-order recurrences. We show that a similar trick exists for Pell-like sequences and does not exist for Jacobhstal-like sequences.