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If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in one geometric progression and $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in another geometric progression, then ap, bq, cr are in
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Geometric progression
Let the common ratio be $\mathrm{K}_{1}$ for $\mathrm{p}, \mathrm{q}$ and $\mathrm{r}$. $\therefore \quad \mathrm{q}=\mathrm{K}_{1} \mathrm{p}$
$\mathrm{\&} \mathrm{r}=\left(\mathrm{K}_{1}\right)^{2} \mathrm{p}$
Let the common ratio be $\mathrm{K}_{2}$ for $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$
$\begin{array}{ll}\therefore & \mathrm{b}=\mathrm{K}_{2} \mathrm{a} \\ \ & \mathrm{c}=\left(\mathrm{K}_{2}\right)^{2} \mathrm{a} \\ \therefore & \mathrm{bq}=\left(\mathrm{K}_{1} \mathrm{~K}_{2}\right) \mathrm{ap} \\ \ & \mathrm{Cr}=\left(\mathrm{K}_{1} \mathrm{~K}_{2}\right)^{2} \text { ap } \\ \text { So } \mathrm{ap}, \mathrm{bq}, \mathrm{cr} \text { are in G.P. }\end{array}$
$\mathrm{\&} \mathrm{r}=\left(\mathrm{K}_{1}\right)^{2} \mathrm{p}$
Let the common ratio be $\mathrm{K}_{2}$ for $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$
$\begin{array}{ll}\therefore & \mathrm{b}=\mathrm{K}_{2} \mathrm{a} \\ \ & \mathrm{c}=\left(\mathrm{K}_{2}\right)^{2} \mathrm{a} \\ \therefore & \mathrm{bq}=\left(\mathrm{K}_{1} \mathrm{~K}_{2}\right) \mathrm{ap} \\ \ & \mathrm{Cr}=\left(\mathrm{K}_{1} \mathrm{~K}_{2}\right)^{2} \text { ap } \\ \text { So } \mathrm{ap}, \mathrm{bq}, \mathrm{cr} \text { are in G.P. }\end{array}$
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