Question:** A sequence is defined by the formula \( a_n = 2n^2 - 3n + 1 \). What is the 10th term of this sequence?
[" discover the 10th Term of the Sequence Defined by ( a_n = 2n^2 - 3n + 1 \ )", "When working with mathematics, particularly sequences, one common question students and learners face is: What is the 10th term of the sequence defined by ( a_n = 2n^2 - 3n + 1 )? This formula represents a quadratic sequence where each term depends on the position ( n ), and understanding how to evaluate it makes mastering sequences much simpler.", "### What is a Sequence?", "A sequence is an ordered list of numbers generated by a specific rule—in this case, the formula ( a_n = 2n^2 - 3n + 1 ). Here, ( n ) is the term number, starting at 1 for the first term.", "### How to Find the 10th Term", "To find the 10th term (( a_{10} )), substitute ( n = 10 ) into the formula:", "[\na_{10} = 2(10)^2 - 3(10) + 1\n]", "Now calculate step-by-step:", "- ( 10^2 = 100 )\n- ( 2 \ imes 100 = 200 )\n- ( 3 \ imes 10 = 30 )\n- Combine: ( 200 - 30 + 1 = 171 )", "Thus, the 10th term is 171.", "### Why This Formula Works", "The expression ( a_n = 2n^2 - 3n + 1 ) defines a quadratic sequence because it includes a term with ( n^2 ). As ( n ) increases, the quadratic term dominates, causing the sequence to grow rapidly. This contrasts with arithmetic sequences, which increase at a constant rate, or linear sequences.", "### How This Formula Can Help You", "Knowing how to evaluate ( a_n ) allows you to:", "- Predict any term in the sequence without listing all previous terms\n- Analyze patterns and derive general properties of the sequence\n- Apply the formula in real-world modeling, physics, computer science, and finance, where quadratic growth is common", "### Final Answer", "The 10th term of the sequence defined by ( a_n = 2n^2 - 3n + 1 ) is:", "[\n\boxed{171}\n]", "If you're learning sequences or facing similar problems, mastering substitution and simplifying polynomial expressions is key. Practice transforming and evaluating formulas to build confidence and fluency in algebra!"]
