Use the method of differences to show that
$$S _ { n } = \frac { 5 n ^ { 2 } + a n } { 12 ( n + b ) ( n + c ) }$$
where \(a , b\) and \(c\) are integers.
Question 14 continues on the next page
14
Show that, for any number \(k\) greater than \(\frac { 12 } { 5 }\), if the difference between \(\frac { 5 } { 12 }\) and \(S _ { n }\) is less than \(\frac { 1 } { k }\), then
$$n > \frac { k - 5 + \sqrt { k ^ { 2 } + 1 } } { 2 }$$