Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }\) in terms of \(n\) and \(k\), where \(k\) is a positive constant.
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It is given that \(\sum _ { r = 1 } ^ { \infty } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) } = \frac { 1 } { 3 }\).
Find the value of \(k\).
Hence find \(\sum _ { r = n } ^ { n ^ { 2 } } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }\) in terms of \(n\).
$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x y$$
has polar equation \(r ^ { 2 } = 3 \sin 2 \theta\).
The curve \(C\) has polar equation \(r ^ { 2 } = 3 \sin 2 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole.
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Find the area of the region enclosed by \(C\).
Find the maximum distance of a point on \(C\) from the initial line.