Problem
ALG-B3-M06-P006 Cauchy on the Integers
#6
★★★☆☆ Level 3 of 5
Find all \(f:\mathbb Z\to\mathbb Z\) such that \(f(m+n)=f(m)+f(n)\) and \(f(1)=-3\).
Use induction and negative integers.
For \(n>0\), \(f(n)=nf(1)=-3n\). Also \(f(0)=0\), and from \(0=f(n+(-n))\), we get \(f(-n)=-f(n)=3n\). Thus for all \(n\in\mathbb Z\), \(f(n)=-3n\).
Standard additivity on \(\mathbb Z\).