A sequence is defined by \( a_n = 3n^2 - 2n + 1 \). Find \( a_5 \).

["Understanding the Sequence Defined by ( a_n = 3n^2 - 2n + 1 ): Finding ( a_5 )", "In mathematics, sequences provide a powerful way to explore patterns and functions. One such sequence is defined by the formula:", "[\na_n = 3n^2 - 2n + 1\n]", "This index-based formula allows us to compute any term ( a_n ) by simply substituting a positive integer value for ( n ). When asked to find ( a_5 ), this means identifying the fifth term in the sequence.", "### What Does ( a_5 ) Represent?", "The notation ( a_5 ) refers to the term obtained when ( n = 5 ) in the sequence. By applying the formula step by step, we can calculate the value precisely.", "### Step-by-Step Calculation of ( a_5 )", "Substitute ( n = 5 ) into the formula:", "[\na_5 = 3(5)^2 - 2(5) + 1\n]", "Now compute each part:", "- ( 5^2 = 25 )\n- ( 3 \ imes 25 = 75 )\n- ( 2 \ imes 5 = 10 )\n- The expression becomes:\n[\na_5 = 75 - 10 + 1\n]", "Simplifying step-by-step:", "[\n75 - 10 = 65\n]\n[\n65 + 1 = 66\n]", "Thus,\n[\na_5 = 66\n]", "### Conclusion", "The fifth term of the sequence defined by ( a_n = 3n^2 - 2n + 1 ) is 66. This demonstrates how straightforward substitution allows quick evaluation of terms in quadratic sequences. Such sequences are foundational in algebra and discrete mathematics, often used in modeling, computer science, and financial calculations.", "If you're studying or working with sequences, mastering this formula and substitution method is essential for solving problems efficiently and accurately. Understanding ( a_n = 3n^2 - 2n + 1 ) and computing ( a_5 = 66 ) serves as a practical example of sequence evaluation.", "---", "Keywords: sequence formula, ( a_n = 3n^2 - 2n + 1 ), find ( a_5 ), evaluation of quadratic sequences, mathematical patterns, algebra practice."]