A sequence is defined by \( a_n = 3n^2 - 2n + 1 \). What is the 10th term?

["Understanding Sequences: Calculating the 10th Term of ( a_n = 3n^2 - 2n + 1 )", "In mathematics, a sequence is an ordered list of numbers defined by a specific rule or formula. One such formula defines a sequence as ( a_n = 3n^2 - 2n + 1 ), where ( n ) represents the term position (a positive integer: 1, 2, 3, ...).", "### What Does the Formula Mean?", "Each term ( a_n ) of the sequence is calculated by plugging the term number ( n ) into the expression:", "[\na_n = 3n^2 - 2n + 1\n]", "This is a quadratic sequence because of the ( n^2 ) term. As ( n ) increases, the value of each term grows rapidly due to the squared component.", "### How to Find the 10th Term", "To find the 10th term, substitute ( n = 10 ) into the formula:", "[\na_{10} = 3(10)^2 - 2(10) + 1\n]", "First, calculate the powers and products:", "[\na_{10} = 3(100) - 20 + 1 = 300 - 20 + 1 = 281\n]", "### Final Result", "The 10th term of the sequence ( a_n = 3n^2 - 2n + 1 ) is:", "[\n\boxed{281}\n]", "### Why Knowing the nth Term Matters", "Understanding how to compute specific terms helps in pattern recognition, solving recurrence relations, and real-world applications like modeling physical phenomena or financial growth. The formula ( a_n = 3n^2 - 2n + 1 ) exemplifies how quadratic growth applies in discrete mathematics.", "If you're studying sequences or exploring patterns, always verify your formula by calculating a few initial terms to ensure accuracy—this builds strong problem-solving skills!"]