Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 4 / 5A / i / Solution

The original PDF contains this subpart; the local TeX has omitted it. Write $a_n$ for the leading coefficient of $P_n$. The Legendre polynomial recurrence relation and the starting polynomials give $a_0=a_1=1$, and induction givesIndeed, $x{P}_n$ has degree of a polynomial $n+1$, while $P_{n-1}$ has smaller degree of a polynomial and cannot cancel its leading term. HenceAt $x=1$, the same linear recurrence relation gives $(n+1)P_{n+1}(1)=(2n+1)P_n(1)-n{P}_{n-1}(1)$. Starting from $P_0(1)=P_1(1)=1$, induction yields