Problem
ALG-B3-M06-P009 The Midpoint Equation on Integers
#9
★★★☆☆ Level 3 of 5
Find all \(f:\mathbb Z\to\mathbb Z\) such that \(f(m+n)+f(m-n)=2f(m)\) for all \(m,n\).
Look at first differences.
Put \(n=1\): \(f(m+1)+f(m-1)=2f(m)\). This means \(f(m+1)-f(m)=f(m)-f(m-1)\), so first differences are constant. Therefore \(f(n)=an+b\). All such functions indeed work.
Discrete version of the midpoint equation.