["Arithmetic Sequence: Exploring the Sequence Defined by (a = 105), (d = 7), and (l = 994)", "Understanding arithmetic sequences is fundamental in mathematics, offering clear patterns and useful applications in various real-world contexts. In this article, we explore a specific arithmetic sequence characterized by its first term (a = 105), common difference (d = 7), and last term (l = 994). We’ll break down its structure, verify key properties, and explain how arithmetic sequences like this one play roles in problem-solving and education.", "---", "### What is an Arithmetic Sequence?", "An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a constant difference (d) to the previous term. Mathematically, the (n)-th term (T_n) of the sequence is given by:", "[
\nT_n = a + (n - 1)d
\n]", "where:
\n- (a) is the first term,
\n- (d) is the common difference,
\n- (n) is the term position.", "---", "### Defining the Given Sequence", "For the sequence described:
\n- First term: (a = 105)
\n- Common difference: (d = 7)
\n- Last term: (l = 994)", "Let’s verify how many terms this sequence contains and confirm that 994 is indeed part of it.", "---", "### Step 1: Find the Number of Terms (n)", "Using the formula for the (n)-th term:", "[
\nT_n = 105 + (n - 1) \ imes 7 = 994
\n]", "Solve for (n):", "[
\n105 + 7(n - 1) = 994
\n]", "Subtract 105 from both sides:", "[
\n7(n - 1) = 889
\n]", "Divide by 7:", "[
\nn - 1 = 127
\n]", "So:", "[
\nn = 128
\n]", "There are 128 terms in this sequence.", "---", "### Step 2: Confirm the Sequence", "Let’s verify both the first and last terms:", "- First term: (T_1 = 105 + (1 - 1) \ imes 7 = 105) ✓
\n- Fifteenth term: (T_{15} = 105 + 14 \ imes 7 = 105 + 98 = 203) ✓
\n- Last term ((n = 128)):
\n [
\n T_{128} = 105 + 127 \ imes 7 = 105 + 889 = 994 \quad \ ext{✓}
\n ]", "The sequence progresses reliably by adding 7 each time.", "---", "### Step 3: Visualizing the Sequence", "Imagine a straight-line diagram where each point represents a term:
\n- The height starts at 105
\n- Each step rises by 7 units", "This linear growth exemplifies the uniform pattern of an arithmetic sequence, making it easy to model in algebra and calculus contexts.", "---", "### Step 4: Applications of Arithmetic Sequences", "Sequences like this one appear in multiple domains:", "- Education: Teaching patterns and number properties
\n- Finance: Calculating interest with fixed periodic additions
\n- Engineering: Scheduling repeatable tasks at consistent intervals
\n- Daily Life: Tracking regular expenses increasing by a fixed amount", "Understanding how to derive and manipulate such sequences equips problem-solvers with a versatile tool.", "---", "### Conclusion", "The arithmetic sequence starting with (a = 105), common difference (d = 7), and last term (l = 994) contains 128 terms, with successive additions of 7 producing a precise final value of 994. Whether for teaching, computation, or modeling real-life scenarios, mastering arithmetic sequences lays a foundational skill in mathematical reasoning.", "If you’re exploring sequences, remember: pattern recognition, formula application, and verification are key—and this sequence is a perfect example to practice and reinforce those skills.", "---", "Keywords: arithmetic sequence, first term formula, common difference, last term of sequence, linear sequence, 128-term sequence, mathematical patterns, algebra in sequences, educational math, disciplinary applications.", "discover more about arithmetic sequences: calculate next terms, compare with geometric sequences, apply in real-world contexts."]
