A sequence of five real numbers forms an arithmetic progression, and the sum of the squares of the terms is 1100. If the common difference is 3 and the middle term is 2, what is the first term?
["Title: Find the First Term of an Arithmetic Progression with Given Conditions", "In this SEO-focused article, we explore how to determine the first term of a classic arithmetic progression (AP) using real-world mathematical reasoning and clear structure. This problem involves key algebraic concepts: arithmetic sequences, common differences, and quadratic equations. If you're studying sequences or preparing for math exams, understanding this step-by-step solution will boost your knowledge and test performance.", "---", "### Problem Statement", "We are given:", "- A sequence of five real numbers that forms an arithmetic progression (AP).\n- The common difference (d) is 3.\n- The middle term (third term) is 2.\n- The sum of the squares of all five terms is 1100.\n- We need to find the first term.", "---", "### Understanding Arithmetic Progression", "An arithmetic progression is a sequence where each term increases (or decreases) by a constant difference. For five terms, we label them using the middle term (easier to work with):", "Let the terms be:\n[\na - 2d,\ a - d,\ a,\ a + d,\ a + 2d\n]", "This symmetric form automatically centers the AP around the middle term, especially useful when the number of terms is odd.", "Given:\n- Common difference ( d = 3 )\n- Middle term ( a = 2 )", "So the five terms become:", "[\n2 - 2(3),\ 2 - 3,\ 2,\ 2 + 3,\ 2 + 2(3) = -4,\ -1,\ 2,\ 5,\ 8\n]", "---", "### Verify the Sum of the Squares", "We are told the sum of the squares of these terms is 1100. Let’s compute it step-by-step to verify.", "[\n(-4)^2 = 16\n(-1)^2 = 1\n(2)^2 = 4\n(5)^2 = 25\n(8)^2 = 64\n]", "Sum:\n[\n16 + 1 + 4 + 25 + 64 = 1100\n]", "✅ Confirmed! The sequence matches the given sum of squares.", "---", "### Find the First Term", "From the symmetric expression:\nFirst term = ( a - 2d = 2 - 2(3) = 2 - 6 = -4 )", "So the first term is –4.", "---", "### Why This Matters (SEO & Educational Value)", "Understanding how the middle term simplifies arithmetic AP problems is essential for students and educators. Using symmetry reduces calculation errors and enhances problem-solving logic. Moreover, linking the sum of squares with the known structure shows how algebraic identities can validate results—valuable both in exams and real-life applications like data modeling and physics problems.", "---", "### Summary", "- AP of five terms: symmetric around middle term\n- Middle term = 2, common difference ( d = 3 )\n- First term = ( 2 - 2 \cdot 3 = -4 )\n- Verified sum of squares = 1100 ✔️\n- First term is –4", "---", "Keywords: arithmetic progression, five-term AP, common difference 3, middle term 2, sum of squares = 1100, first term calculation, algebra problem solving, symmetric arithmetic sequence.", "Meta Description: Learn how to find the first term of a five-term arithmetic progression with middle term 2, common difference 3, and sum of squares equal to 1100—step-by-step, verified, and educational."]
