Let `z_2 = 1 + i` and, for `n > 2`, let `z_n = z_(n - 1) (1 + i/(|\ z_(n - 1)\ |)).`
Use mathematical induction to prove that `|\ z_n\ | = sqrt n` for all integers `n >= 2.` (3 marks)
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Let `z_2 = 1 + i` and, for `n > 2`, let `z_n = z_(n - 1) (1 + i/(|\ z_(n - 1)\ |)).`
Use mathematical induction to prove that `|\ z_n\ | = sqrt n` for all integers `n >= 2.` (3 marks)
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`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Prove)\ \ |\ z_n\ | = sqrt n\ \ \ text(for)\ n>=2`
`text(given)\ \ z_n = z_(n – 1) (1 + i/(|\ z_(n – 1)\ |))`
`text(If)\ \ n=2`
` z_2 = 1 + i`
`|\ z_2\ | = sqrt (1 + 1) = sqrt 2`
`:.text(True for)\ \ n = 2`
`text(Assume that)\ \ |\ z_k\ | = sqrt k\ \ text(and prove that)\ \ |\ z_(k + 1)\ | = sqrt (k + 1)`
| `z_k` | `= z_(k – 1) (1 + i/(|\ z_(k – 1)\ |))` |
| `=> z_(k + 1) ` | `= z_k (1 + i/(|\ z_k\ |))` |
| `|\ z_(k + 1)\ |` | `= |\ z_k (1 + i/(|\ z_k\ |))\ |` |
| `=|\ z_k\ |*|\ (1 + i/sqrt k)\ |` | |
| `= sqrt k * sqrt((1^2+(1/sqrt k)^2))` | |
| `=sqrt k * sqrt((1+1/k))` | |
| `= sqrt k * sqrt((k+1)/k)` | |
| `= sqrt k * sqrt(k+1)/sqrt k` | |
| `=sqrt(k+1)` |
`=>text(True for)\ \ n = k + 1`
`:. text(S)text(ince true for)\ \ n=2,\ \ text(by PMI, true for all integers)\ \ n >= 2.`