Problem
ALG-B1-M11-P016 Polynomial difference
#16
★★★★☆ Level 4 of 5
A polynomial \(P\) satisfies \(P(x+1)-P(x)=2x+1\), \(P(0)=0\). Find \(P(n)\) for integers \(n\ge0\).
Sum the equalities from \(0\) to \(n-1\).
\(P(n)-P(0)=\sum_{k=0}^{n-1}(2k+1)=n^2\). Hence \(P(n)=n^2\).
Strategy: polynomial differences as a telescoping sum.