Three numbers a,b,c are between 2 and 18 such that their sum is 25 and 2,a,b are A.P and b,c,18 are in G.P then abc  is 

$$\begin{gathered} given that a+b+c=25 \to \left(1\right) \\ \sin ce 2,a,b are in A.P \Rightarrow \end{gathered}$$

$\begin{array}{l}2+\mathrm{b}=2\mathrm{a}----\left(2\right)\\ \mathrm{b},\mathrm{c},18\text{ are in }\mathrm{G}\cdot \mathrm{P}\Rightarrow 18\mathrm{b}={\mathrm{c}}^2-\cdots -\left(3\right)\\ \left(1\right)\text{ and}\left(2\right)\text{, eliminate a }\Rightarrow 3\mathrm{b}+2\mathrm{c}=48----\left(4\right)\\ \left(3\right)\text{ and }\left(4\right)\Rightarrow {\mathrm{c}}^2+12c=288\to \mathrm{c}=12,-24\\ \to \mathrm{c}\in (2,18)\Rightarrow \mathrm{c}=12,\mathrm{b}=8,\mathrm{a}=5\to \mathrm{abc}=480\end{array}$