A sequence is defined by \( a_n = 3n^2 - 2n + 1 \). Find \( a_4 \).
["Understanding Sequences: Calculating ( a_4 ) for ( a_n = 3n^2 - 2n + 1 )", "In mathematics, sequences provide a powerful way to explore patterns and relationships through ordered lists of numbers. One common method to define such sequences is using a closed-form formula that expresses the ( n )-th term directly. In this article, we explore the sequential definition ( a_n = 3n^2 - 2n + 1 ) and specifically compute ( a_4 ), the fourth term in the sequence.", "---", "### What Is a Sequence?", "A sequence is an ordered list of numbers indexed by natural numbers (usually ( n = 1, 2, 3, \ldots )). For example, if ( a_n = 3n^2 - 2n + 1 ), then each term is generated by plugging in a positive integer ( n ).", "---", "### The Formula: ( a_n = 3n^2 - 2n + 1 )", "This expression defines the ( n )-th term of the sequence. By substituting any positive integer ( n ), we can calculate the corresponding value. The formula combines a quadratic term (( 3n^2 )), a linear term (( -2n )), and a constant (( +1 )), creating a non-linear but predictable pattern.", "---", "### How to Compute ( a_4 )", "To find the fourth term in the sequence, substitute ( n = 4 ) into the formula:", "[\na_4 = 3(4)^2 - 2(4) + 1\n]", "Now compute step by step:", "- ( 4^2 = 16 )\n- ( 3 \ imes 16 = 48 )\n- ( 2 \ imes 4 = 8 )\n- Now combine: ( 48 - 8 + 1 = 41 )", "So,", "[\na_4 = 41\n]", "---", "### Why Knowing ( a_4 ) Matters", "Calculating individual terms like ( a_4 ) helps in pattern recognition, problem-solving, and modeling real-world situations where sequences represent growth patterns, financial projections, or data trends.", "---", "### Summary", "The sequence defined by ( a_n = 3n^2 - 2n + 1 ) delivers a well-structured series of values. For the fourth term:", "[\na_4 = 41\n]", "Mastering such expressions enables students, educators, and professionals to analyze numerical patterns with precision and insight.", "---", "### Further Reading", "- Explore other general quadratic sequences and pattern recognition techniques\n- Learn how to derive closed-form formulas from recursive definitions\n- Study applications of sequences in computer science, finance, and physics", "---", "Keywords: sequence definition, ( a_n = 3n^2 - 2n + 1 ), find ( a_4 ), mathematical formulas, closed-form expression, quadratic sequence."]
