- (a) Show that for \(r \geqslant 1\)
$$\frac { r } { \sqrt { r ( r + 1 ) } + \sqrt { r ( r - 1 ) } } \equiv A ( \sqrt { r ( r + 1 ) } - \sqrt { r ( r - 1 ) } )$$
where \(A\) is a constant to be determined.
(b) Hence use the method of differences to determine a simplified expression for
$$\sum _ { r = 1 } ^ { n } \frac { r } { \sqrt { r ( r + 1 ) } + \sqrt { r ( r - 1 ) } }$$
(c) Determine, as a surd in simplest form, the constant \(k\) such that
$$\sum _ { r = 1 } ^ { n } \frac { k r } { \sqrt { r ( r + 1 ) } + \sqrt { r ( r - 1 ) } } = \sqrt { \sum _ { r = 1 } ^ { n } r }$$