3 The irrational number \(\phi = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\) plays a significant role in the sequence of Fibonacci numbers given by \(\mathrm { F } _ { 0 } = 0 , \mathrm {~F} _ { 1 } = 1\) and \(\mathrm { F } _ { \mathrm { n } + 1 } = \mathrm { F } _ { \mathrm { n } } + \mathrm { F } _ { \mathrm { n } - 1 }\) for \(n \geqslant 1\). Prove by induction that, for each positive integer \(n , \phi ^ { n } = \mathrm { F } _ { \mathrm { n } } \times \phi + \mathrm { F } _ { \mathrm { n } - 1 }\).