7 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
$$u _ { 1 } = 2 , \quad u _ { k + 1 } = 2 u _ { k } + 1$$
- Prove by induction that, for all \(n \geqslant 1\),
$$u _ { n } = 3 \times 2 ^ { n - 1 } - 1$$
- Show that
$$\sum _ { r = 1 } ^ { n } u _ { r } = u _ { n + 1 } - ( n + 2 )$$