Problem:** The sum of the first \( n \) terms of an arithmetic sequence is \( S_n = 3n^2 + 5n \). What is the 10th term?
["Title: How to Find the 10th Term of an Arithmetic Sequence Given the Sum Formula", "Meta Description:\nDiscover how to find the 10th term of an arithmetic sequence when the sum of the first ( n ) terms is given by ( S_n = 3n^2 + 5n ). Learn step-by-step how to derive the formula and calculate the term.", "---", "### Understanding the Sum of an Arithmetic Sequence", "An arithmetic sequence is a sequence where each term increases by a constant difference ( d ). The sum of the first ( n ) terms, denoted ( S_n ), is given by the formula:", "[\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, the sum is given as a quadratic expression:", "[\nS_n = 3n^2 + 5n\n]", "This suggests a faster-growing sequence, consistent with the nature of arithmetic sequences where the partial sums grow quadratically.", "---", "### Step 1: Find a General Expression for the ( n )-th Term", "The ( n )-th term ( a_n ) of an arithmetic sequence can also be found using the sum formula:", "[\na_n = S_n - S_{n-1}\n]", "So, compute ( S_{n-1} ) by substituting ( n-1 ) into the given sum:", "[\nS_{n-1} = 3(n-1)^2 + 5(n-1)\n]\n[\nS_{n-1} = 3(n^2 - 2n + 1) + 5n - 5 = 3n^2 - 6n + 3 + 5n - 5 = 3n^2 - n - 2\n]", "Now compute the ( n )-th term:", "[\na_n = S_n - S_{n-1} = (3n^2 + 5n) - (3n^2 - n - 2) = 3n^2 + 5n - 3n^2 + n + 2 = 6n + 2\n]", "So, the general term is:", "[\na_n = 6n + 2\n]", "---", "### Step 2: Find the 10th Term", "Use the formula to find the 10th term:", "[\na_{10} = 6(10) + 2 = 60 + 2 = 62\n]", "---", "### Summary", "Given that the sum of the first ( n ) terms of an arithmetic sequence is ( S_n = 3n^2 + 5n ), the 10th term is:", "[\n\boxed{62}\n]", "---", "### Why This Method Works for Arithmetic Sequences", "Because the sum grows quadratically, the underlying sequence must be arithmetic — each term increases linearly. By computing ( S_n - S_{n-1} ), we isolate the ( n )-th term directly, revealing linear behavior even when only the sum formula is known.", "---", "Keywords: arithmetic sequence, sum of first n terms, find nth term, formula derivation, ( S_n = 3n^2 + 5n ), ( a_{10} ), mathematical sequences, quadratic sum sequence.", "---", "Read also: How to Derive Arithmetic Sequence Formulas from Sum Expressions, Find Any Term in a Quadratic Sum Sequence, Arithmetic Sequence Problems Explained."]
