Show that
$$\frac { 2 ^ { r + 1 } } { r + 2 } - \frac { 2 ^ { r } } { r + 1 } = \frac { r 2 ^ { r } } { ( r + 1 ) ( r + 2 ) }$$
Hence find
$$\sum _ { r = 1 } ^ { 30 } \frac { r 2 ^ { r } } { ( r + 1 ) ( r + 2 ) }$$
giving your answer in the form \(2 ^ { n } - 1\), where \(n\) is an integer.