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

["Understanding the Sum of an Arithmetic Sequence: Finding the 10th Term Given ( S_n = 3n^2 + 5n )", "The sum of the first ( n ) terms of an arithmetic sequence is given by the formula:", "[\nS_n = \frac{n}{2} \left(2a + (n-1)d\right)\n]", "However, in this case, we are provided with a different formula:", "[\nS_n = 3n^2 + 5n\n]", "This allows us to directly analyze the sequence and compute specific terms without needing the explicit values of the first term ( a ) and common difference ( d ).", "### The Relationship Between ( S_n ) and Individual Terms", "For any arithmetic sequence, the sum of the first ( n ) terms can also be expressed as:", "[\nS_n = S_{n-1} + a_n\n]", "This means the ( n )th term ( a_n ) is:", "[\na_n = S_n - S_{n-1}\n]", "Using the given formula ( S_n = 3n^2 + 5n ), we 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 compute ( a_n ):", "[\na_n = S_n - S_{n-1} = (3n^2 + 5n) - (3n^2 - n - 2) = 3n^2 + 5n - 3n^2 + n + 2 = 6n + 2\n]", "### Finding the 10th Term", "Using the derived formula for the ( n )th term:", "[\na_n = 6n + 2\n]", "Substitute ( n = 10 ):", "[\na_{10} = 6 \cdot 10 + 2 = 60 + 2 = 62\n]", "### Conclusion", "The sum formula ( S_n = 3n^2 + 5n ) reveals that the ( n )th term of the arithmetic sequence is linear: ( a_n = 6n + 2 ). Using this, the 10th term is:", "[\n\boxed{62}\n]", "This approach demonstrates how modern sequence sum formulas simplify finding individual terms without needing to extract the first term or difference explicitly. Whether analyzing patterns or solving advanced problems, recognizing such relationships is key in mathematics and data modeling contexts."]