Coefficients of a cubic in terms of its roots, and the constant term when the roots are in arithmetic or geometric progression

The roots of the cubic equation \(x^3+ax^2+bx+k=0\) are \(\alpha\), \(\beta\) and \(\gamma\). Expand \((x-\alpha)(x-\beta)(x-\gamma)\) and compare coefficients to show that

1. 1. \(a=-(\alpha+\beta+\gamma)\)
   2. \(b=\alpha\beta+\beta\gamma+\gamma\alpha\)
   3. \(k=-\alpha\beta\gamma\).[3]

In the rest of this question, \(a=-9\) and \(b=36\).

2. 1. Suppose that \(\alpha\), \(\beta\) and \(\gamma\) are consecutive terms of an arithmetic sequence. Show that \(3\) is a root of the equation.
   2. Hence find \(k\).[5]
3. Suppose instead that \(\alpha\), \(\beta\) and \(\gamma\) are consecutive terms of a geometric sequence. Find \(k\).[6]