If $\mathrm{p}^{\text {th }}, \mathrm{q}^{\text {th }}$ and $\mathrm{r}^{\text {th }}$ terms of H.P. are $\mathrm{u}, \mathrm{v}, \mathrm{w}$ respectively, then find the value of the expression $(q-r) v w+(r-p) w u+(p-q) u v$.

Question

If $\mathrm{p}^{\text {th }}, \mathrm{q}^{\text {th }}$ and $\mathrm{r}^{\text {th }}$ terms of H.P. are $\mathrm{u}, \mathrm{v}, \mathrm{w}$ respectively, then find the value of the expression $(q-r) v w+(r-p) w u+(p-q) u v$., 2, 0, 4, 8

MARKS App · Accepted Answer

Let H.P. be $\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{a}+\mathrm{d}}+\frac{1}{\mathrm{a}+2 \mathrm{~d}}+\ldots \ldots$ $\begin{aligned} & \therefore \mathrm{u}=\frac{1}{\mathrm{a}+(\mathrm{p}-1) \mathrm{d}}, \mathrm{v}=\frac{1}{\mathrm{a}+(\mathrm{q}-1) \mathrm{d}}, \ & \mathrm{w}=\frac{1}{\mathrm{a}+(\mathrm{r}-1) \mathrm{d}} \Rightarrow \mathrm{a}+(\mathrm{p}-1) \mathrm{d}=\frac{1}{\mathrm{u}} \ & \mathrm{a}+(\mathrm{q}-1) \mathrm{d}=\frac{1}{\mathrm{v}}, \mathrm{a}+(\mathrm{r}-1) \mathrm{d}=\frac{1}{\mathrm{w}} \ & \Rightarrow(\mathrm{q}-\mathrm{r})\{\mathrm{a}+(\mathrm{p}-1) \mathrm{d}\}+(\mathrm{r}-\mathrm{p})\{\mathrm{a}+(\mathrm{q}-1) \mathrm{d}\}+\ldots \ldots \ & =\frac{1}{\mathrm{u}}(\mathrm{q}-\mathrm{r})+\frac{1}{\mathrm{v}}(\mathrm{r}-\mathrm{p})+\ldots . \Rightarrow(q-r) \mathrm{vw}+\ldots . . = 0 \end{aligned}$

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