3 (a) Use the method of differences to find $\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }$ in terms of $n$ and $k$, where $k$ is a positive constant.\\

(b) Deduce the value of $\sum _ { \mathrm { r } = 1 } ^ { \infty } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }$.\\

(c) Find also $\sum _ { \mathrm { r } = \mathrm { n } } ^ { \mathrm { n } ^ { 2 } } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }$ in terms of $n$ and $k$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q3}}