The sum of the first \( n \) terms of an arithmetic sequence is given by \( S_n = 2n^2 + 3n \). Find the 10th term of the sequence.

["The Sum of the First ( n ) Terms Formula ( S_n = 2n^2 + 3n ) and How to Find the 10th Term", "Understanding the sum of terms in an arithmetic sequence is fundamental to mastering sequences and series. Given the formula for the sum of the first ( n ) terms:\n[ S_n = 2n^2 + 3n ]\nthis article explains how to derive key properties of the sequence—especially how to find the 10th term—using algebraic techniques rooted in arithmetic progression concepts.", "---", "### How to Find the ( n^{\ ext{th}} ) Term from the Sum Formula", "In any arithmetic sequence, the ( n^{\ ext{th}} ) term, denoted ( a_n ), can be expressed using the sum formula:\n[ a_n = S_n - S_{n-1} ]", "Start by computing ( S_{n-1} ):\n[ S_{n-1} = 2(n-1)^2 + 3(n-1) ]\n[ = 2(n^2 - 2n + 1) + 3n - 3 ]\n[ = 2n^2 - 4n + 2 + 3n - 3 ]\n[ = 2n^2 - n - 1 ]", "Now subtract:\n[\na_n = S_n - S_{n-1} = (2n^2 + 3n) - (2n^2 - n - 1)\n= 2n^2 + 3n - 2n^2 + n + 1\n= 4n + 1\n]", "Thus, the ( n^{\ ext{th}} ) term of the sequence is:\n[\na_n = 4n + 1\n]", "---", "### Applying the Formula to Find the 10th Term", "Substitute ( n = 10 ) into ( a_n = 4n + 1 ):\n[\na_{10} = 4(10) + 1 = 40 + 1 = 41\n]", "---", "### Verification: Confirm the Formula Works", "Since ( S_n = 2n^2 + 3n ) describes a quadratic sum, the original sequence is actually quadratic, typical of arithmetic sequences where the common difference ( d ) is constant.", "We know:\n- The standard sum formula for an arithmetic sequence is:\n[\nS_n = \frac{n}{2}(2a + (n - 1)d)\n]\nExpanding this:\n[\nS_n = \frac{d}{2}n^2 + \left( \frac{2a - d}{2} \right)n\n]\nCompare with ( S_n = 2n^2 + 3n ):\n[\n\frac{d}{2} = 2 \Rightarrow d = 4\n]\n[\n\frac{2a - 4}{2} = 3 \Rightarrow 2a - 4 = 6 \Rightarrow a = 5\n]\nSo the first term ( a = 5 ), common difference ( d = 4 ). The sequence starts:\n[\n5, 9, 13, 17, \dots\n]\nNow compute ( a_{10} = 5 + (10 - 1)\cdot 4 = 5 + 36 = 41 ), confirming our earlier result.", "---", "### Why This Method Matters", "This approach—finding ( a_n = S_n - S_{n-1} )—is powerful for arbitrary summation formulas, especially when sequences grow quadratically. It emphasizes the deep connection between cumulative sums and individual term values, a cornerstone concept in series and sequences.", "---", "### Conclusion", "Given the sum of the first ( n ) terms of an arithmetic sequence is ( S_n = 2n^2 + 3n ), the ( n^{\ ext{th}} ) term is simply ( a_n = 4n + 1 ). Therefore, the 10th term is:\n[\n\boxed{41}\n]", "Understanding such formulas not only helps solve specific problems but strengthens analytical skills essential in mathematics, physics, and engineering."]