A sequence is defined by \( a_n = 3n^2 - 2n + 1 \). Find the 5th term.
["Understanding Sequences: Finding the 5th Term of ( a_n = 3n^2 - 2n + 1 )", "In mathematics, sequences help us study patterns and relationships through ordered lists of numbers. One powerful way to define a sequence is using a formula that specifies the ( n )-th term explicitly—this is known as a closed-form expression. An excellent example is the quadratic sequence defined by:", "[\na_n = 3n^2 - 2n + 1\n]", "Here, ( a_n ) represents the ( n )-th term of the sequence, where ( n ) is a positive integer (1, 2, 3, ...).", "### How to Find the 5th Term", "To determine the 5th term in this sequence, substitute ( n = 5 ) directly into the formula:", "[\na_5 = 3(5)^2 - 2(5) + 1\n]", "Break it down step by step:", "- First, calculate ( 5^2 = 25 )\n- Multiply: ( 3 \ imes 25 = 75 )\n- Next, compute ( 2 \ imes 5 = 10 )\n- Now combine: ( 75 - 10 + 1 = 66 )", "Thus, the 5th term is:", "[\n\boxed{66}\n]", "### Why Knowing the nth Term Matters", "Using formulas like this simplifies finding any term without calculating all preceding ones. This efficiency is crucial in science, engineering, finance, and computer science, where pattern recognition and prediction depend heavily on algebraic expressions.", "### Conclusion", "The sequence defined by ( a_n = 3n^2 - 2n + 1 ) is rich with contained information. By plugging in ( n = 5 ), we quickly determine that ( a_5 = 66 ). Whether for learning, problem-solving, or real-world applications, understanding how to extract terms from formulas like this strengthens analytical skills and deepens mathematical intuition."]
