The sum of the first \( n \) terms of an arithmetic sequence is \( S_n = 3n^2 + 5n \). Find the 10th term.
["## Finding the 10th Term of an Arithmetic Sequence Given the Sum Formula", "Understanding the formula for the sum of the first ( n ) terms of an arithmetic sequence is crucial for solving many mathematical problems, especially when direct term inspection is required. In this article, we explore how to deduce the 10th term of an arithmetic sequence when the sum of the first ( n ) terms is given by:", "[\nS_n = 3n^2 + 5n\n]", "### The Sum Formula and Arithmetic Sequences", "For any arithmetic sequence, the sum of the first ( n ) terms can be expressed as:", "[\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 direct quadratic formula for ( S_n ), which allows us to deduce the underlying sequence without explicitly identifying ( a ) and ( d ).", "### Step 1: Relate ( S_n ) to Individual Terms", "The ( n )th term, ( a_n ), can be found using the relationship:", "[\na_n = S_n - S_{n-1}\n]", "This formula lets us compute any specific term directly from the sum function.", "### Step 2: Compute ( S_n ) and ( S_{n-1} )", "Given ( S_n = 3n^2 + 5n ), 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]", "### Step 3: Find the ( n )th Term", "Now, subtract to find ( 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]", "Thus, the ( n )th term of the sequence is:", "[\na_n = 6n + 2\n]", "### Step 4: Calculate the 10th Term", "Substitute ( n = 10 ):", "[\na_{10} = 6(10) + 2 = 60 + 2 = 62\n]", "### Conclusion", "By leveraging the given sum formula ( S_n = 3n^2 + 5n ), we derived the explicit formula for the ( n )th term and calculated the 10th term efficiently. This method avoids the need to reverse-engineer ( a ) and ( d ) explicitly, showcasing how sum formulas reveal key sequence properties.", "Answer: The 10th term of the sequence is ( \boxed{62} )."]
