3 The sequence \(\left\{ U _ { n } \right\}\) is given by \(U _ { 1 } = 0 , U _ { 2 } = - 1\) and \(U _ { n + 2 } = U _ { n + 1 } + U _ { n } + n - 1\) for \(n \geqslant 1\).
- List the first seven terms of this sequence.
The Fibonacci sequence \(\left\{ \mathrm { F } _ { \mathrm { n } } \right\}\) is given by \(\mathrm { F } _ { 1 } = 1 , \mathrm {~F} _ { 2 } = 1\) and \(\mathrm { F } _ { \mathrm { n } + 2 } = \mathrm { F } _ { \mathrm { n } + 1 } + \mathrm { F } _ { \mathrm { n } }\) for \(n \geqslant 1\).
- By comparing the two sequences, give the relationship between \(\mathrm { U } _ { \mathrm { n } }\) and \(\mathrm { F } _ { \mathrm { n } }\).
- Show that the relationship found in part (b)(i) holds for all \(n \geqslant 1\).