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 and Finding the 10th Term of an Arithmetic Sequence Using the Sum Formula", "The sum of the first ( n ) terms of an arithmetic sequence is a fundamental concept in algebra, widely applied in mathematical analysis, finance, and data science. Given the sum formula ( S_n = 3n^2 + 5n ), this article explains how to deduce the 10th term of the underlying arithmetic sequence—without requiring explicit knowledge of the first term or common difference.", "---", "### The Sum of an Arithmetic Sequence: Recap", "For any arithmetic sequence, the sum of the first ( n ) terms is given by:", "[\nS_n = \frac{n}{2} \left(2a + (n - 1)d\right)\n]", "where ( a ) is the first term and ( d ) is the common difference.", "However, in this problem, we are given a closed-form expression:", "[\nS_n = 3n^2 + 5n\n]", "This quadratic expression suggests a direct relationship between the number of terms and their cumulative sum—perfect for backward inference of individual terms.", "---", "### Deriving the General Term ( a_n ) from ( S_n )", "The ( n )th term of the sequence can be computed using the identity:", "[\na_n = S_n - S_{n-1}\n]", "This is because ( S_n - S_{n-1} ) gives the sum of all terms from ( a_1 ) to ( a_n ), minus the sum up to ( a_{n-1} ), isolating the ( n )th term.", "Let’s compute ( S_{n-1} ):", "[\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 subtract:", "[\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 general term is:", "[\na_n = 6n + 2\n]", "---", "### Finding the 10th Term", "Substitute ( n = 10 ) into the formula:", "[\na_{10} = 6(10) + 2 = 60 + 2 = 62\n]", "---", "### Verification: Compute ( S_{10} ) and Confirm Consistency", "Using the given sum formula:", "[\nS_{10} = 3(10)^2 + 5(10) = 300 + 50 = 350\n]", "Now compute the sum of the first 10 terms using the derived term formula ( a_n = 6n + 2 ). This is an arithmetic sequence (since ( a_n ) is linear), so:", "[\nS_n = \frac{n}{2}(a_1 + a_n)\n]", "First, find ( a_1 ) and ( a_{10} ):", "[\na_1 = 6(1) + 2 = 8,\quad a_{10} = 62\n]", "[\nS_{10} = \frac{10}{2}(8 + 62) = 5 \ imes 70 = 350\n]", "Matches the earlier result—confirming consistency.", "---", "### Conclusion", "Given the sum formula ( S_n = 3n^2 + 5n ), we successfully derived the general term and computed the 10th term as ( a_{10} = 62 ). This approach—using differences of partial sums—provides a powerful, elegant method to uncover individual terms in arithmetic sequences without prior knowledge of ( a ) or ( d ).", "Using such formulas enables efficient modeling in real-world applications like interest calculations, structured data growth, and predictive analytics.", "---", "Keywords: arithmetic sequence sum formula, ( S_n = 3n^2 + 5n ), finding the 10th term, general term of arithmetic sequence, mathematical derivation, algebra, student resources, educational math."]