Given that \(z = 22 + 4 \mathrm { i }\) and \(\frac { z } { w } = 6 - 8 \mathrm { i }\), find
\(w\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real,
the argument of \(z\), in radians to 2 decimal places.
(a) Prove that \(\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r - 1 ) = \frac { 1 } { 6 } n ( n - 1 ) ( 2 n + 5 )\).
Deduce that \(n ( n - 1 ) ( 2 n + 5 )\) is divisible by 6 for all \(n > 1\). [0pt]
[P4 January 2002 Qn 3]
$$\mathrm { f } ( x ) = x ^ { 3 } + x - 3$$
The equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), between 1 and 2 .
By considering \(\mathrm { f } ^ { \prime } ( x )\), show that \(\alpha\) is the only real root of the equation \(\mathrm { f } ( x ) = 0\).
Taking 1.2 as your first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 significant figures.
Prove that your answer to part (b) gives the value of \(\alpha\) correct to 3 significant figures.