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If the sum of n terms of an A.P. is 3$\mathbf{n}^{2}$ + 5n then which of its terms is 164 ?

A. 26th B. 27th C. 28th D. None of these Answer: Option B
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Solution(By Apex Team)

Sum of n terms of an A.P. = $3 n^{2}+5 n$ Let a be the first term and d be the common difference $\begin{array}{l}\mathrm{S}_{\mathrm{n}}=3\mathrm{n}^2+5\mathrm{n}\\ \mathrm{S}_1=3(1)^2+5\times1=3+5=8\\ \mathrm{~S}_2=3(2)^2+5\times2=12+10=22\\ \therefore\text{ First term }(\mathrm{a})=8\\ \mathrm{a}_2=\mathrm{S}_2-\mathrm{S}_1=22-8=14\\ \mathrm{~d}=\mathrm{a}_2-\mathrm{a}_1=14-8=6\\ \text{ Now }\mathrm{a}_{\mathrm{n}}=\mathrm{a}+(\mathrm{n}-1)\mathrm{d}\\ \Rightarrow164=8+(\mathrm{n}-1)\times6\\ \Rightarrow6\mathrm{n}-6=164-8\\ \Rightarrow6\mathrm{n}=156+6\\ \Rightarrow6\mathrm{n}=162\end{array}$ ⇒ n = $\Large\frac{162}{6}$ ⇒ n = 27 ∴ 168 is 27th term

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