Problem
ALG-B3-M06-P005 Second Differences
#5
★★☆☆☆ Level 2 of 5
Let \(f:\mathbb Z\to\mathbb Z\), \(f(0)=0\), \(f(1)=2\), and \(f(n+2)-2f(n+1)+f(n)=0\) for all \(n\). Find \(f(n)\).
Show that first differences are constant.
The condition is equivalent to \(f(n+2)-f(n+1)=f(n+1)-f(n)\). Thus first differences are constant. Since \(f(1)-f(0)=2\), we have \(f(n+1)-f(n)=2\) for all \(n\). Hence \(f(n)=2n\).
Discrete analogue of linearity.