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Q.
If $l, m , n$ are respectively the $p ^{th}, q^{th}, r^{th}$ terms of an $H.P$. then$\begin{vmatrix}mn &ln &lm\\ p &q &r\\ 1&1&1\end{vmatrix}$ equals to
Sequences and Series
Solution:
Let the first term of an $A.P$. be $A$ and common difference is $d$.
As $l, m, n$ are terms of $H.P$.
$\therefore \frac{1}{l}, \frac{1}{m}, \frac{1}{n}$ are terms of $A.P$.
$\therefore \frac{1}{l} =A +\left(p -1\right)d$
$\therefore mn \left(q-1\right)= lmn\left[A +\left(p -1\right)d\right]\left(q -r\right)$
(Multiplying & dividing by $I$)
$\frac{1}{m} =A +\left(q -1\right)d$
$\therefore ln\left(r-p\right) =lmn\left[A+\left(q-1\right)d\right]\left(r -p\right)$
$\frac{1}{n} =A +\left(r-1\right)d$
$\therefore lm\left(p-q\right) = lmn \left[A+\left(r -1\right)d\right]\left(p -q\right)$
Adding the above relations, we get
$mn\left(q -r\right) +ln \left(r -p\right) +lm \left(p-q\right) =0$
or $\begin{vmatrix}mn &ln &lm\\ p &q &r\\ 1&1&1\end{vmatrix}=0$