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If the sequence $\left\{\mathrm{S}_{\mathrm{n}}\right\}$ is a geometric progression and $\mathrm{S}_{2} \mathrm{~S}_{11}=\mathrm{S}_{\mathrm{p}} \mathrm{S}_{8}$, then what is the value of $\mathrm{p}$ ?
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The correct answer is:
5
Given $\mathrm{S}_{2} \mathrm{~S}_{11}=\mathrm{S}_{\mathrm{p}} \mathrm{S}_{8}$
$\Rightarrow($ ar $)\left(\mathrm{ar}^{10}\right)=\mathrm{ar}^{\mathrm{P}-1}\left(\mathrm{ar}^{7}\right)$
where 'a' is the first term and ' $\mathrm{r}$ ' is the common ratio of GP. $\Rightarrow r^{11}=r^{7+p-1} \Rightarrow r^{11}=r^{6+p}$
$\Rightarrow 11=6+\mathrm{p} \Rightarrow \mathrm{p}=5$
$\Rightarrow($ ar $)\left(\mathrm{ar}^{10}\right)=\mathrm{ar}^{\mathrm{P}-1}\left(\mathrm{ar}^{7}\right)$
where 'a' is the first term and ' $\mathrm{r}$ ' is the common ratio of GP. $\Rightarrow r^{11}=r^{7+p-1} \Rightarrow r^{11}=r^{6+p}$
$\Rightarrow 11=6+\mathrm{p} \Rightarrow \mathrm{p}=5$
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