Consider $9$ terms $a_1,a_2 \cdots a_9$ in Harmonic Progression with $a_4=5,a_5=4$. Find the value of the determinant $$\begin{vmatrix}a_1&a_2&a_3\\a_4&5&4\\ a_7&a_8&a_9\end{vmatrix}$$ The 'not so good ' method is clear, calculating the terms and then evaluating, but I want to ask if there exists a beautiful or elegant solution. Something I am missing. Thanks.
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$a_1(5a_9 - 4a_8) - a_2*(a_4a_9-4a_7)+a_3(a_4a_8-5a_7)$ – Cardinal Jul 18 '15 at 16:06
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Yeah, this is the 'not so good' method. The calculation is highly tedious. Please give some elegant solution. – Jul 18 '15 at 16:08
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2What does H.P mean? – Empy2 Jul 18 '15 at 16:08
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Did you mean $\begin{vmatrix}a_1&a_2&a_3\5&4&a_6\ a_7&a_8&a_9\end{vmatrix}$ ? Else $a_1 = \infty$. – Marconius Jul 18 '15 at 20:04