Question 3 - A-Level Maths

Express $( k + 1 ) ^ { 2 } + 5 ( k + 1 ) + 8$ in the form $k ^ { 2 } + a k + b$, where $a$ and $b$ are constants.
Prove by induction that, for all integers $n \geqslant 1$, $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) \left( \frac { 1 } { 2 } \right) ^ { r - 1 } = 16 - \left( n ^ { 2 } + 5 n + 8 \right) \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$$