Problem
ALG-B3-M02-P004 Difference of arguments
#4
★★★☆☆ Level 3 of 5
Let \(f:\mathbb Z\to\mathbb Z\), \(f(x-y)=f(x)-f(y)\), and \(f(2)=10\). Find \(f(n)\).
Hint. First find \(f(0)\), then \(f(1)\).
With \(x=y\), \(f(0)=0\). With \(x=2\), \(y=1\): \(f(1)=f(2)-f(1)\), so \(f(1)=5\). The equation gives additivity on \(\mathbb Z\), hence \(f(n)=5n\).
Goal: choose the right first substitution and then verify the found function.