$\begin{align}  & {{U}_{n}}=a{{r}^{n-1}} \\ & {{U}_{2}}=ar=24----(i) \\ & {{U}_{5}}=a{{r}^{4}}=81---(ii) \\ & \frac{{{U}_{5}}}{{{U}_{2}}}=\frac{a{{r}^{4}}}{ar}=\frac{81}{24} \\ & {{r}^{3}}=\frac{27}{8} \\ & r=\frac{3}{2} \\ & \text{Substitute }r=\frac{3}{2}\text{ in equation }(i) \\ & \frac{3a}{2}=24 \\ & a=\frac{24\times 2}{3}=16 \\ & {{U}_{7}}=a{{r}^{6}}=16\times {{\left( \frac{3}{2} \right)}^{6}}=\frac{729}{4} \\\end{align}$