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If the sum of n terms of an A.P. be 3$\mathbf{n}^{2}$ + n and its common difference is 6, then its first term is.

A. 2 B. 3 C. 1 D. 4 Answer: Option D
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Solution(By Apex Team)

Sum of n terms of an A.P. = 3$n^{2}$ + n and common difference (d) = 6 Let first term be a, then $\begin{aligned}&\therefore S_n=\frac{n}{2}[2a+(n-1)d]=3n^2+n\\ &\Rightarrow\frac{n}{2}[2a+(n-1)6]=3n^2+n\\ &\Rightarrow2a+6n-6=\left(3n^2+n\right)\times\frac{2}{n}\\ &\Rightarrow2a+6n-6=n\frac{(3n+1)\times2}{n}\\ &\Rightarrow2a+6n-6=(3n+1)2\\ &\Rightarrow2a+6n-6=6n+2\\ &\Rightarrow2a=6n+2-6n+6\\ &\Rightarrow2a=8\\ &\therefore a=\frac{8}{2}\\ &=4\end{aligned}$

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