\(n = 8\), \(a = 5\), \(d = 3\).
["Understanding the Arithmetic Sequence Formula: (n = 8), (a = 5), (d = 3)", "In mathematics, arithmetic sequences are foundational for modeling patterns and solving real-world problems. Defined by a starting term and a common difference, these sequences follow a simple yet powerful formula. Today, we explore a specific arithmetic sequence—where the number of terms (n = 8), the first term (a = 5), and the common difference (d = 3)—and demonstrate its structure, key properties, and how to compute its terms efficiently.", "---", "### What Is an Arithmetic Sequence?", "An arithmetic sequence is a sequence of numbers in which each term increases (or decreases) by a constant amount, known as the common difference (d). The general formula for the (n)-th term (n) is:", "[\na_n = a + (n - 1)d\n]", "- (a_n): value of the (n)-th term\n- (a): first term of the sequence\n- (d): common difference between consecutive terms\n- (n): position (term number) of the term", "---", "### Applying the Formula to (n = 8), (a = 5), (d = 3)", "Given:\n- First term (a = 5)\n- Common difference (d = 3)\n- Number of terms (n = 8)", "We compute the sequence step-by-step:", "- Term 1 ((n=1)): (a_1 = 5 + (1 - 1) \ imes 3 = 5)\n- Term 2 ((n=2)): (a_2 = 5 + (2 - 1) \ imes 3 = 8)\n- Term 3 ((n=3)): (a_3 = 5 + (3 - 1) \ imes 3 = 11)\n- Term 4 ((n=4)): (a_4 = 5 + (4 - 1) \ imes 3 = 14)\n- Term 5 ((n=5)): (a_5 = 5 + (5 - 1) \ imes 3 = 17)\n- Term 6 ((n=6)): (a_6 = 5 + (6 - 1) \ imes 3 = 20)\n- Term 7 ((n=7)): (a_7 = 5 + (7 - 1) \ imes 3 = 23)\n- Term 8 ((n=8)): (a_8 = 5 + (8 - 1) \ imes 3 = 26)", "So, the full sequence is:\n5, 8, 11, 14, 17, 20, 23, 26", "---", "### Key Observations", "- First term is clearly (a = 5).\n- The spread between terms is consistent: each term increases by (d = 3).\n- The 8th term (a_8) is calculated as (5 + (8 - 1) \ imes 3 = 26), confirming the location and value.", "---", "### Real-World Applications", "This sequence models scenarios with steady growth or decline. Examples include:", "- Daily savings: Saving $5 daily, with an extra $3 bonus per day increases total savings predictably.\n- Stair clipping: Each step on a stair rises 3 inches; counting 8 steps at 5-inch base height.\n- Thermal scaling: Temperature rising by 3°C per hour starting from 5°C over 8 hours.", "---", "### Bonus: Sum of the First 8 Terms", "To find the total, compute the sum (S_n) using the formula:\n[\nS_n = \frac{n}{2}(a_1 + a_n)\n]\nSubstitute values:\n[\nS_8 = \frac{8}{2}(5 + 26) = 4 \ imes 31 = 124\n]", "Alternatively:\n[\nS_8 = \frac{8}{2}[2 \ imes 5 + (8 - 1) \ imes 3] = 4[10 + 21] = 4 \ imes 31 = 124\n]", "This sum represents cumulative values over eight periods with consistent growth.", "---", "### Conclusion", "The sequence defined by (n = 8), (a = 5), (d = 3) illustrates how arithmetic progressions simplify modeling and calculation. Whether used in finance, science, or everyday planning, recognizing these patterns enhances problem-solving skills. Remember: locate the first term, apply the constant difference, and leverage formulas efficiently. The 8-term sequence with (a=5), (d=3) produces values from 5 to 26, and its total sum is 124, demonstrating the elegance of arithmetic sequences.", "---", "Keywords: arithmetic sequence, (n = 8), (a = 5), (d = 3), arithmetic progression, linear sequence, math formula, sequence calculation, real-world examples, sum of arithmetic sequence."]
