Problem
ALG-B1-M10-P021 Function on integers
#21
★★★★☆ Level 4 of 5
Let \(f:\mathbb Z\to\mathbb Z\), \(f(m+n)=f(m)+f(n)+2mn\), \(f(1)=2\). Find \(f(n)\).
Subtract \(n^2\).
Let \(g(n)=f(n)-n^2\). Then \(g(m+n)=g(m)+g(n)\), and \(g(1)=1\). Hence \(g(n)=n\), so \(f(n)=n^2+n\).
A functional equation with a quadratic extra term.