the value of \(d\);

the value of \(r\);

EITHER

the first three terms of the geometric sequence are \(9\), \(9r\) and \(9{r^2}\)     (M1)

\(9 + 3d = 9r( \Rightarrow 3 + d = 3r)\) and \(9 + 7d = 9{r^2}\)     (A1)

attempt to solve simultaneously     (M1)

\(9 + 7d = 9{\left( {\frac{{3 + d}}{3}} \right)^2}\)

OR

the \({{\text{1}}^{{\text{st}}}}\), \({{\text{4}}^{{\text{th}}}}\) and \({{\text{8}}^{{\text{th}}}}\) terms of the arithmetic sequence are

\(9,{\text{ }}9 + 3d,{\text{ }}9 + 7d\)     (M1)

\(\frac{{9 + 7d}}{{9 + 3d}} = \frac{{9 + 3d}}{9}\)     (A1)

attempt to solve     (M1)

THEN

\(d = 1\)     A1

[4 marks]

\(r = \frac{4}{3}\)     A1

Note:     Accept answers where a candidate obtains \(d\) by finding \(r\) first. The first two marks in either method for part (a) are awarded for the same ideas and the third mark is awarded for attempting to solve an equation in \(r\).

[1 mark]