These form an arithmetic sequence: \(108, 120, \dots, 996\) with \(a = 108\), \(l = 996\), \(d = 12\).
["# Understanding the Arithmetic Sequence: (108, 120, \dots, 996) with (a = 108), (d = 12), (l = 996)", "An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by (d). Exploring arithmetic sequences helps build foundational mathematical reasoning used in algebra, number theory, and real-world applications.", "## The Given Sequence\nThe sequence under consideration starts at (a = 108) and increases by (d = 12) each step, ending at (l = 996). This means the sequence follows the pattern:\n[ 108,\ 120,\ \dots,\ 996 ]", "## Defining the Key Parameters\nTo analyze the full sequence, let’s define the essential arithmetic sequence parameters:\n- First term ((a)): 108\n- Common difference ((d)): 12\n- Last term ((l)): 996", "## Formula for the (n)-th Term\nThe general formula for the (n)-th term ((a_n)) of an arithmetic sequence is:\n[\na_n = a + (n - 1)d\n]\nSubstituting the known values:\n[\na_n = 108 + (n - 1) \cdot 12\n]", "## Finding the Number of Terms ((n))\nWe are given that the last term (l = 996). Setting (a_n = 996), we solve for (n):\n[\n996 = 108 + (n - 1) \cdot 12\n]\nSubtract 108 from both sides:\n[\n888 = (n - 1) \cdot 12\n]\nDivide both sides by 12:\n[\nn - 1 = 74\n]\n[\nn = 75\n]\nSo, there are 75 terms in this arithmetic sequence.", "## Verifying the Last Term\nCheck the last term using (n = 75):\n[\na_{75} = 108 + (75 - 1) \cdot 12 = 108 + 74 \cdot 12 = 108 + 888 = 996\n]\nConfirmed — the sequence correctly ends at 996.", "## Visualizing the Sequence\nThis sequence starts at 108, adds 12 repeatedly, and progresses as:\n108, 120, 132, 144, ..., 984, 996\nThe consistent difference of 12 ensures each term is precisely 12 more than the previous.", "## Applications of Arithmetic Sequences\nArithmetic sequences appear in diverse contexts:\n- Financial calculations (e.g., fixed periodic savings)\n- Scheduling tasks at regular intervals\n- Mathematical modeling of evenly spaced phenomena\nUnderstanding such sequences strengthens problem-solving skills and supports algebraic development.", "---", "Summary\nThe arithmetic sequence (108, 120, \dots, 996) with first term (a = 108), common difference (d = 12), and last term (l = 996) contains 75 terms. Using the formula (a_n = a + (n - 1)d), we derived (n = 75), confirming the sequence’s length and validity. This structured approach enables efficient analysis of similar sequences in mathematics and applied fields."]
