Problem
ALG-B1-M08-P008 Squares from differences
#8
★★☆☆☆ Level 2 of 5
Let \(f(0)=0\) and \(f(n+1)=f(n)+2n+1\) for all integers \(n\ge0\). Prove that \(f(n)=n^2\) for all \(n\ge0\).
Add the odd numbers \(1,3,\ldots,2n-1\).
Adding the equalities from \(0\) to \(n-1\), we get \(f(n)-f(0)=1+3+\cdots+(2n-1)=n^2\). Since \(f(0)=0\), \(f(n)=n^2\).
This is a functional notation for the familiar sum of odd numbers.