If ${\mathrm{a}}_1, {\mathrm{a}}_2, {\mathrm{a}}_3, 5, 4, {\mathrm{a}}_6, {\mathrm{a}}_7, {\mathrm{a}}_8, {\mathrm{a}}_9$ are in H.P., and $\mathrm{D}=\left|\begin{array}{ccc}{\mathrm{a}}_1& {\mathrm{a}}_2& {\mathrm{a}}_3\\ 5& 4& {\mathrm{a}}_6\\ {\mathrm{a}}_7& {\mathrm{a}}_8& {\mathrm{a}}_9\end{array}\right|$, then the value of [D] is (where [.] represents the greatest integer function) __________ .

We have $\mathrm{D}=\left|\begin{array}{ccc}{\mathrm{a}}_1& {\mathrm{a}}_2& {\mathrm{a}}_3\\ 5& 4& {\mathrm{a}}_6\\ {\mathrm{a}}_7& {\mathrm{a}}_8& {\mathrm{a}}_9\end{array}\right|$

Since ${\mathrm{a}}_{\mathrm{n}}=\frac{20}{\mathrm{n}};\mathrm{d}=\frac{1}{20}$

Hence, $\mathrm{D}=\left|\begin{array}{ccc}20& \frac{20}{2}& \frac{20}{3}\\ \frac{20}{4}& \frac{20}{5}& \frac{20}{6}\\ \frac{20}{7}& \frac{20}{8}& \frac{20}{9}\end{array}\right|=\frac{(20{)}^3}{4\times 7}\left|\begin{array}{ccc}1& \frac{{\displaystyle 1}}{{\displaystyle 2}}& \frac{{\displaystyle 1}}{{\displaystyle 3}}\\ 1& \frac{{\displaystyle 4}}{{\displaystyle 5}}& \frac{{\displaystyle 2}}{{\displaystyle 3}}\\ 1& \frac{{\displaystyle 7}}{{\displaystyle 8}}& \frac{{\displaystyle 7}}{{\displaystyle 9}}\end{array}\right|$

${\mathrm{R}}_1\to {\mathrm{R}}_1-{\mathrm{R}}_2\text{ and }{\mathrm{R}}_2\to {\mathrm{R}}_2-{\mathrm{R}}_3$

$=\frac{(20{)}^3}{4\times 7}\left|\begin{array}{ccc}0& \frac{-3}{10}& \frac{-1}{3}\\ 0& \frac{-3}{40}& \frac{-1}{9}\\ 1& \frac{7}{8}& \frac{7}{9}\end{array}\right|=\frac{50}{21}$

$\Rightarrow \left[\mathrm{D}\right]=2$