Substitute \( n = 4 \): \( a_4 = 3(4)^2 - 2(4) + 1 = 48 - 8 + 1 = 41 \).

["# Substitute ( n = 4 ) in the Sequence Formula: ( a_4 = 3(4)^2 - 2(4) + 1 = 41 )", "When exploring arithmetic or recursive sequences, one common method is to substitute specific values of ( n ) into the general formula to compute individual terms. This article dives into the case where ( n = 4 ) in the quadratic recurrence defined by the formula:", "[\na_n = 3n^2 - 2n + 1\n]", "## What Is This Sequence?", "At the heart of this example is a second-degree polynomial sequence:\n[\na_n = 3n^2 - 2n + 1\n]\nThis type of sequence frequently appears in discrete mathematics, computer science, and algebra problems. It generates terms that grow quadratically, meaning the difference between consecutive terms increases linearly.", "## Calculating ( a_4 ): Step-by-Step", "To find ( a_4 ), substitute ( n = 4 ) directly into the formula:", "[\na_4 = 3(4)^2 - 2(4) + 1\n]", "Break it down term by term:\n- First term: ( 3 \ imes (4)^2 = 3 \ imes 16 = 48 )\n- Second term: ( -2 \ imes 4 = -8 )\n- Constant term: ( +1 )", "Now sum the results:\n[\n48 - 8 + 1 = 41\n]", "Thus,\n[\na_4 = 41\n]", "## Why Understanding ( n = 4 ) Matters", "Computing ( a_4 ) is more than a simple substitution—it helps in understanding how quadratic sequences behave:", "- Pattern recognition: By calculating terms for ( n = 1, 2, 3, 4 ), students or learners can identify patterns in how the sequence develops.\n- Error checking: When implementing the formula in code or verifying mathematical models, early terms like ( a_4 = 41 ) serve as key benchmarks.\n- Foundations for recurrence relations: Quadratic sequences like this often arise in recurrence relations and calculus of finite differences, making ( a_4 ) a useful evaluation point.", "## Expanding Your Sequence Knowledge", "To deepen understanding, consider comparing ( a_n ) with adjacent terms:", "| ( n ) | ( 3n^2 ) | ( -2n ) | +1 | Total = ( a_n ) |\n|--------|-----------|----------|----|------------------|\n| 3 | 27 | -6 | +1 | 23 |\n| 4 | 48 | -8 | +1 | 41 |\n| 5 | 75 | -10 | +1 | 66 |", "From this breakdown, observe that the sequence increases steadily, with each term depending on the quadratic growth minus a linear adjustment, stabilized by the constant.", "## Conclusion", "Substituting ( n = 4 ) into the formula ( a_n = 3n^2 - 2n + 1 ) yields\n[\na_4 = 3(4)^2 - 2(4) + 1 = 48 - 8 + 1 = 41\n]\nThis simple yet revealing calculation underscores the clarity of quadratic sequences and serves as a building block for more complex recursive or closed-form formulations. Whether in classroom learning, programming exercises, or mathematical problem-solving, mastering such substitutions strengthens analytical skills and computational fluency.", "---", "Keywords: substitute ( n = 4 ), sequence formula ( a_n = 3n^2 - 2n + 1 ), ( a_4 = 41 ), quadratic sequence, polynomial evaluation, mathematical calculation, discrete mathematics."]