6 Prove by induction that $$\frac { 2 \times 1 } { 2 \times 3 } + \frac { 2 ^ { 2 } \times 2 } { 3 \times 4 } + \frac { 2 ^ { 3 } \times 3 } { 4 \times 5 } + \ldots + \frac { 2 ^ { n } \times n } { ( n + 1 ) ( n + 2 ) } = \frac { 2 ^ { n + 1 } } { n + 2 } - 1$$ for all integers \(n \geqslant 1\).