The sum of the first \(n\) terms of an arithmetic sequence is given by \(S_n = 3n^2 + 5n\). Find the 10th term of the sequence.

["Understanding the Sum Formula to Find the 10th Term of an Arithmetic Sequence", "The sum of the first (n) terms of an arithmetic sequence—denoted as (S_n)—plays a fundamental role in understanding sequence behavior and calculating specific term values efficiently. In this article, we explore how the given formula (S_n = 3n^2 + 5n) reveals both the structure of the sequence and the exact value of its 10th term.", "---", "### The Standard Formula for Arithmetic Sequences", "Recall that for any arithmetic sequence, the sum of the first (n) terms is normally given by:", "[\nS_n = \frac{n}{2}(2a + (n-1)d)\n]", "where (a) is the first term and (d) is the common difference.", "However, in this problem, we are given a quadratic expression for (S_n = 3n^2 + 5n), which directly links the sum formula to a degree second in (n). This allows us to bypass expanding the standard sum formula and instead derive term values through (\Delta_n = S_n - S_{n-1}), the difference between consecutive sums.", "---", "### Deriving the (n)th Term Using the Summation Formula", "The (n)th term (a_n) of an arithmetic sequence is the difference between successive sums:", "[\na_n = S_n - S_{n-1}\n]", "Substitute (S_n = 3n^2 + 5n) and compute:", "[\nS_{n-1} = 3(n-1)^2 + 5(n-1) = 3(n^2 - 2n + 1) + 5n - 5 = 3n^2 - 6n + 3 + 5n - 5 = 3n^2 - n - 2\n]", "Now calculate:", "[\na_n = S_n - S_{n-1} = (3n^2 + 5n) - (3n^2 - n - 2) = 3n^2 + 5n - 3n^2 + n + 2 = 6n + 2\n]", "Thus, the (n)th term of the sequence is:", "[\na_n = 6n + 2\n]", "---", "### Finding the 10th Term", "To find the 10th term ((a_{10})), substitute (n = 10) into the derived formula:", "[\na_{10} = 6(10) + 2 = 60 + 2 = 62\n]", "---", "### Verifying via Direct Use of (S_n)", "For additional clarity, compute (S_{10}) and (S_9) using the original formula:", "[\nS_{10} = 3(10)^2 + 5(10) = 300 + 50 = 350\n]\n[\nS_9 = 3(9)^2 + 5(9) = 243 + 45 = 288\n]\n[\na_{10} = S_{10} - S_9 = 350 - 288 = 62\n]", "Same result confirms accuracy.", "---", "### Conclusion", "When the sum of the first (n) terms is a quadratic expression like (S_n = 3n^2 + 5n), the sequence is linear, and the (n)th term can be found efficiently via difference methods or direct substitution into the derived formula. Here, the 10th term of the arithmetic sequence is:", "[\n\boxed{62}\n]", "Understanding this approach enables quick analysis of sequences with polynomial sum formulas—key for mastering discrete mathematics and algebraic modeling.", "---", "Keywords: arithmetic sequence sum formula, find 10th term, derive nth term, summation of arithmetic terms, quadratic sum (S_n = 3n^2 + 5n), arithmetic progression, mathematical sequences."]