होम Quiz Post If Sn denotes the sum of the first r terms of.

If Sn denotes the sum of the first r terms of.

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If $\mathbf{S}_{\mathbf{n}}$ denotes the sum of the first r terms of an A.P. Then, $\mathbf{S}_{3 \mathrm{n}}:\left(\mathbf{S}_{2 \mathrm{n}}-\mathbf{S}_{\mathbf{n}}\right)$ is

A. n B. 3n C. 3 D. None of these Answer: Option C
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Solution(By Apex Team)

$\begin{aligned}S_n&=\frac{n}{2}[2a+(n-1)d]\\ S_{2n}&=\frac{2n}{2}[2a+(2n-1)d]\text{ and }\\ S_{3n}&=\frac{3n}{2}[2a+(3n-1)d]\\ &\text{ Now }S_{2n}-S_n&\\ &=\frac{2n}{2}[2a+(2n-1)d]-\frac{n}{2}[2a+(n-1)d]\\ &=\frac{n}{2}[4a+(4n-2)d]-[2a+(n-1)d]\\ &=\frac{n}{2}[4a-2a+(4n-2-n+1)d]\\ &=\frac{n}{2}[2a+(3n-1)d]\\ &=\frac{1}{3}\left(S_{3n}\right)\\ &\therefore S_{3n}:\left(S_{2n}-S_n\right)\\ &=3:1\text{ or }\frac{3}{1}=3\end{aligned}$

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