Solution: Let the four terms be $ a - \frac{3d}{2},\ a - \frac{d}{2},\ a + \frac{d}{2},\ a + \frac{3d}{2} $. This ensures the terms are in arithmetic progression with common difference $ d $. The first term is $ a - \frac{3d}{2} $, and the last term is $ a + \frac{3d}{2} $. Their sum of squares is:
["Solution: Understanding the Sum of Squares in Arithmetic Progression", "When analyzing sequences in algebra, identifying structure and symmetry can simplify seemingly complex expressions—especially when dealing with arithmetic progressions (APs). Consider the four terms in arithmetic progression:", "$$\na - \frac{3d}{2},\quad a - \frac{d}{2},\quad a + \frac{d}{2},\quad a + \frac{3d}{2}\n$$", "These four terms form a clean arithmetic sequence with common difference $ d $. Starting from $ a - \frac{3d}{2} $ and ending at $ a + \frac{3d}{2} $, they are evenly spaced, making their sum of squares particularly manageable.", "Let’s compute the sum of squares of these four terms step by step.", "---", "### Step 1: Write the expression for the sum of squares", "[\n\left(a - \frac{3d}{2}\right)^2 + \left(a - \frac{d}{2}\right)^2 + \left(a + \frac{d}{2}\right)^2 + \left(a + \frac{3d}{2}\right)^2\n]", "---", "### Step 2: Expand each square", "- First term:\n $$ \left(a - \frac{3d}{2}\right)^2 = a^2 - 3ad + \frac{9d^2}{4} $$", "- Second term:\n $$ \left(a - \frac{d}{2}\right)^2 = a^2 - ad + \frac{d^2}{4} $$", "- Third term:\n $$ \left(a + \frac{d}{2}\right)^2 = a^2 + ad + \frac{d^2}{4} $$", "- Fourth term:\n $$ \left(a + \frac{3d}{2}\right)^2 = a^2 + 3ad + \frac{9d^2}{4} $$", "---", "### Step 3: Add all expanded expressions", "Now sum them:", "[\n\begin{align}\n& (a^2 - 3ad + \ frac{9d^2}{4}) + (a^2 - ad + \ frac{d^2}{4}) \\n& + (a^2 + ad + \ frac{d^2}{4}) + (a^2 + 3ad + \ frac{9d^2}{4}) \\n= &\ a^2 + a^2 + a^2 + a^2 \\n& + (-3ad - ad + ad + 3ad) \\n& + \left(\ frac{9d^2}{4} + \ frac{d^2}{4} + \ frac{d^2}{4} + \ frac{9d^2}{4}\right) \\n= &\ 4a^2 + 0\cdot ad + \left( \ frac{9 + 1 + 1 + 9}{4} d^2 \right) \\n= &\ 4a^2 + \frac{20d^2}{4} \\n= &\ 4a^2 + 5d^2\n\end{align}\n]", "---", "### Final Result:", "$$\n\boxed{4a^2 + 5d^2}\n$$", "---", "### Why This Insight Matters", "This symmetric structure allows us to quickly compute the sum of squares without expanding and combining 16 square terms individually. Recognizing that the set is centered at $ a $ with intervals spaced by $ d $ halves the computational effort—ideal when solving for optimization, minimizing error, or analyzing polynomial sequences in algebra and calculus.", "Whether you're teaching APs, solving polynomial identities, or preparing for advanced math competitions, this method exemplifies how symmetry and structure simplify algebra.", "---", "Keywords: arithmetic progression sum of squares, algebraic simplification, sum of squares formula, mathematical derivation, symmetric sequences, calculus prep, algebra tips, solve algebra problems, symmetric terms, polynomial expansion."]
