A sequence is defined by \(a_n = 3n^2 - 2n + 1\). Find the 10th term.
["Understanding Sequences: How to Find the 10th Term Using the Formula (a_n = 3n^2 - 2n + 1)", "In mathematics, sequences are ordered lists of numbers that follow specific patterns defined by formulas. One such sequence is defined by the general term:\n[ a_n = 3n^2 - 2n + 1 ]\nwhere ( n ) represents the term position in the sequence (a positive integer), and ( a_n ) is the value of the ( n )-th term.", "### What Does the Formula Mean?", "The formula ( a_n = 3n^2 - 2n + 1 ) allows us to compute any term directly by substituting the value of ( n ). For example:\n- To find the 1st term, plug in ( n = 1 ):\n ( a_1 = 3(1)^2 - 2(1) + 1 = 3 - 2 + 1 = 2 )\n- To find the 2nd term, ( a_2 = 3(2)^2 - 2(2) + 1 = 12 - 4 + 1 = 9 ), and so on.", "This explicit formula makes it easy to compute terms without waiting for previous values.", "### Calculating the 10th Term", "To find the 10th term, substitute ( n = 10 ) into the formula:\n[\na_{10} = 3(10)^2 - 2(10) + 1\n]\nStep-by-step calculation:\n- ( 10^2 = 100 )\n- Multiply: ( 3 \ imes 100 = 300 )\n- Subtract: ( 2 \ imes 10 = 20 ), so ( 300 - 20 = 280 )\n- Add: ( 280 + 1 = 281 )", "Thus,\n[ a_{10} = 281 ]", "### Why Understand Sequence Formulas?", "Knowing how to compute terms in sequences is essential in algebra, calculus, computer science, and engineering. Especially for quadratic sequences like this one, where each term depends on a polynomial expression, mastering the formula saves time and enhances problem-solving skills.", "### Summary", "- The sequence is defined by ( a_n = 3n^2 - 2n + 1 ) for ( n = 1, 2, 3, \dots )\n- The 10th term is calculated as ( a_{10} = 3(10)^2 - 2(10) + 1 = 281 )\n- Direct substitution from the formula avoids iteration and simplifies computation", "If you're studying sequences or learning polynomial patterns in math, practice computing terms using explicit formulas — it’s a clear, efficient approach!", "---", "Keywords: sequence definition, ( a_n = 3n^2 - 2n + 1 ), find ( a_{10} ), exponential and polynomial sequences, teach math formulas, algebraic formulas, quadratic sequence."]
