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If in an A.P., $\mathbf{S}_{\mathbf{n}}=\mathbf{n}^{2} \mathbf{p}$ and $\mathbf{S}_{\mathbf{m}}=\mathbf{m}^{\mathbf{2}} \mathbf{p}$, where S denotes the sum of r terms of the A.P., then $\mathbf{S}_{\mathbf{p}}$ is equal to

A. $\frac{1}{2} p^{3}$ B. mnp C. $\mathrm{p}^{3}$ D. $(m+n) p^{2}$ Answer: Option C
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Solution(By Apex Team)

$\begin{array}{l}\mathrm{S}_n=\mathrm{n}^2\mathrm{p},\mathrm{S}_{\mathrm{m}}=\mathrm{m}^2\mathrm{p}\\ \therefore\mathrm{S}_{\mathrm{r}}=\mathrm{r}^2\mathrm{p}\text{ and }\mathrm{S}_{\mathrm{p}}=\mathrm{p}^2\mathrm{p}=\mathrm{p}^3\\ \text{ Hence, }\mathrm{S}_{\mathrm{p}}=\mathrm{p}^3\end{array}$

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