Quadratic factor (difference of squares)

1. (a) Express \(\frac { 1 } { 4 r ^ { 2 } - 1 }\) in partial fractions.
   (b) Hence prove that

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$ (c) Find the exact value of $$\sum _ { r = 9 } ^ { 25 } \frac { 5 } { 4 r ^ { 2 } - 1 }$$

1. (a) Express \(\frac { 2 } { r \left( r ^ { 2 } - 1 \right) }\) in partial fractions.
   (b) Hence find, in terms of \(n\),

$$\sum _ { r = 2 } ^ { n } \frac { 1 } { r \left( r ^ { 2 } - 1 \right) }$$ Give your answer in the form $$\frac { n ^ { 2 } + A n + B } { C n ( n + 1 ) }$$ where \(A\), \(B\) and \(C\) are constants to be found.