The sum of the first $n$ terms of an arithmetic progression is given by $S_n = 3n^2 + 5n$. The 10th term ($T_{10}$) of this arithmetic progression is:

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### 1. Strategy & Theoretical Principles For any sequence where the sum of $n$ terms is $S_n$, the $n$-th term is given by: $$T_n = S_n - S_{n-1}$$ Alternatively, differentiate or evaluate directly at $n = 10$ and $n = 9$. ### 2. Analytical Formulation & Calculation Given $S_n = 3n^2 + 5n$: - For $n = 10$: $$S_{10} = 3(10)^2 + 5(10) = 3(100) + 50 = 350$$ - For $n = 9$: $$S_9 = 3(9)^2 + 5(9) = 3(81) + 45 = 243 + 45 = 288$$ Evaluating $T_{10}$: $$T_{10} = S_{10} - S_9 = 350 - 288 = 62$$ Alternatively, $T_n = S_n - S_{n-1} = 3n^2 + 5n - [3(n-1)^2 + 5(n-1)] = 6n + 2$. At $n = 10$: $$T_{10} = 6(10) + 2 = 62$$ ### 3. Distractor Forensics - Candidate evaluating $S_{10} / 10 = 35$ reports the average, not the 10th term. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Always use $T_n = S_n - S_{n-1}$ for sum formulas. - **Standard Reference:** Aggarwal, R.S., *Quantitative Aptitude*.

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