2020 SJI P1 Q7

A boy claims to have found a function that only produces prime numbers. He then writes the function as

\[\pi(n)=n^2+n+5,\quad n\in\mathbb Z^+.\]

Prove or disprove his claim.[3]

A girl claims to have discovered a formula that reads

\[\sum_{r=1}^n\frac{r}{(r+1)!}=1-\frac1{(n+1)!},\quad n\in\mathbb Z^+.\]

Prove her discovery using mathematical induction.[7]