A sequence is defined by the formula a_n = 3n + 2. What is the 10th term?
["Understanding Linear Sequences: The Formula aₙ = 3n + 2 and Its 10th Term", "In mathematics, sequences are fundamental tools used in algebra, calculus, computer science, and problem-solving across disciplines. One common type of sequence is a linear sequence defined by a clear formula. A classic example is the arithmetic sequence given by the formula:", "aₙ = 3n + 2", "In this article, we will explore what this formula defines, why it represents a sequence, and how to compute key terms—specifically, the 10th term.", "---", "### What Is a Sequence Defined by a Formula?", "A sequence is an ordered list of numbers where each term follows a specific pattern or rule. When defined by a formula like aₙ = 3n + 2, the position of the term, n, determines its value. Here, n is the term index starting at 1, and aₙ is the value of the term at that position.", "Because n increases by 1 with each term, and the formula is linear (i.e., involves n with exponent 1), this sequence is an arithmetic sequence—a sequence with a constant difference between consecutive terms.", "---", "### Breaking Down the Formula: aₙ = 3n + 2", "The formula aₙ = 3n + 2 tells us how to calculate any term in the sequence:", "- n = term number (starting at 1)\n- Multiply n by 3\n- Add 2", "For example, plugging in n = 1 gives the first term:\na₁ = 3(1) + 2 = 5\nSimilarly, a₂ = 3(2) + 2 = 8, and so on.", "This linear relationship ensures the sequence grows steadily, increasing by 3 each time.", "---", "### How to Find the 10th Term", "To find the 10th term ((a_{10})), simply substitute n = 10 into the formula:", "[\na_{10} = 3(10) + 2 = 30 + 2 = 32\n]", "So, the 10th term is 32.", "---", "### Why Knowing the 10th Term Matters", "Understanding the 10th term isn’t just about computing a number—it’s a gateway to deeper insights:", "- Pattern recognition: The sequence progresses consistently: 5, 8, 11, 14, ..., showing a pattern of adding 3.\n- Problem-solving applications: This sequence can model real-world scenarios, such as constant growth over time (e.g., saving money at fixed intervals).\n- Foundation for advanced math: Mastery of linear sequences supports algebra, graphing linear functions, and understanding recursive relations.", "---", "### Summary", "- The sequence defined by aₙ = 3n + 2 is a linear arithmetic sequence.\n- Each term is generated by multiplying its position n by 3 and adding 2.\n- To find any term, use the formula directly. For the 10th term:\na₁₀ = 3(10) + 2 = 32\n- Recognizing such formulas helps in analyzing progressions, solving equations, and modeling real-life phenomena.", "Whether you’re a student learning sequences for the first time or a self-learner exploring pattern-based math, understanding formulas like aₙ = 3n + 2 builds essential analytical skills with clear, practical value.", "---", "Want to practice? Try computing the 20th term or explore how changing the formula (e.g., doubling the coefficient to aₙ = 5n + 4) affects the sequence. Formula-based sequences are powerful tools—keep exploring!"]
