1 Let $a$ be a positive constant.\\
(a) Use the method of differences to find $\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { ( \mathrm { ar } + 1 ) ( \mathrm { ar } + \mathrm { a } + 1 ) }$ in terms of $n$ and $a$.\\

(b) Find the value of $a$ for which $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( a r + 1 ) ( a r + a + 1 ) } = \frac { 1 } { 6 }$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q1}}