Problem
ALG-B3-M05-P002 Additivity on Integers
#2
★★☆☆☆ Level 2 of 5
Let \(f:\mathbb Z\to\mathbb Z\), \(f(a+b)=f(a)+f(b)\), and \(f(2)=10\). Find \(f(n)\).
First find \(f(1)\), then negative arguments.
From \(f(2)=f(1)+f(1)\), we get \(f(1)=5\). For positive \(n\), \(f(n)=5n\). Also \(0=f(0)=f(n+(-n))=f(n)+f(-n)\), so \(f(-n)=-5n\). The answer is \(f(n)=5n\) for all \(n\in\mathbb Z\).
Checks handling of negative integers.