1 (a) By considering $( r + 1 ) ^ { 2 } - r ^ { 2 }$, use the method of differences to prove that

$$\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )$$

(b) Given that $\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( \mathrm { r } + \mathrm { a } ) = \mathrm { n }$, find $a$ in terms of $n$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q1}}