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]