On integral consecutive arithmetic means of the first Fibonacci numbers

Abstract

In this note, we prove a conjecture of Fatehizadeh and Yaqubi regarding the arithmetic mean of the first n Fibonacci numbers. More precisely, we show that there are infinitely many positive integers n such that \(n \mid \sum _{i=1}^{n} F_i\) and \(n+1 \mid \sum _{i=1}^{n+1} F_i\).