Question 6 - A-Level Maths

Show that $\frac { 1 } { ( k + 2 ) ! } - \frac { k + 1 } { ( k + 3 ) ! } = \frac { 2 } { ( k + 3 ) ! }$.
Prove by induction that, for all positive integers $n$, $$\sum _ { r = 1 } ^ { n } \frac { r \times 2 ^ { r } } { ( r + 2 ) ! } = 1 - \frac { 2 ^ { n + 1 } } { ( n + 2 ) ! }$$ (6 marks)
\includegraphics[max width=\textwidth, alt={}]{d417bc62-f92a-4c90-a4c2-435f38e46edc-7_2010_1711_693_152}