Question: An ornithologist observes that the number of birds returning to a nesting site each year follows a sequence: $ a, a+d, a+2d, a+3d, a+4d $, forming an arithmetic sequence. If the sum of the squares of these five terms is 225 and the middle term is 5, find the value of $ a $.

["An ornithologist observes that the number of birds returning to a nesting site each year follows a sequence: $ a, a+d, a+2d, a+3d, a+4d $, forming an arithmetic sequence. If the sum of the squares of these five terms is 225 and the middle term is 5, find the value of $ a $.", "A Surprising Pattern in Nature’s Cycles \nBird nesting patterns reveal quiet order hidden in data—one year’s arrival numbers forming a predictable arithmetic rhythm. Using a single central observation—the 3rd year’s return count—we unlock a richer understanding of seasonal consistency and mathematical precision in wildlife behavior. This sequence is key to understanding population stability and migration trends.", "Why This Pattern Matters in Ornithology Today \nAcross the U.S., ornithologists track yearly cycles to assess environmental health and species resilience. An arithmetic sequence suggests environmental stability—clouded by short-term fluctuations like weather or food availability. Recent data trends show increasing awareness of long-term ecological patterns, aligning with public interest in climate impacts and biodiversity preservation. Understanding these sequences helps scientists and enthusiasts alike interpret biodiversity data with confidence.", "How the Sequence Unfolds \nDefining the five terms: \n- Year 1: $ a $ \n- Year 2: $ a + d $ \n- Year 3: $ a + 2d $ ← known maximum: $ a + 2d = 5 $ \n- Year 4: $ a + 3d $ \n- Year 5: $ a + 4d $", "Since the middle term is 5 and this is the third term, $ a + 2d = 5 $. This center-point gives a foundational anchor, simplifying the sum of squares calculation.", "Solving with Math and Context \nStart with the known: $ a + 2d = 5 $ → $ a = 5 - 2d $ \nNow compute each term’s square: \n- $ (a)^2 = (5 - 2d)^2 $ \n- $ (a + d)^2 = (5 - d)^2 $ \n- $ (a + 2d)^2 = 5^2 = 25 $ \n- $ (a + 3d)^2 = (5 + d)^2 $ \n- $ (a + 4d)^2 = (5 + 2d)^2 $", "Sum these: \n\[\n(5 - 2d)^2 + (5 - d)^2 + 25 + (5 + d)^2 + (5 + 2d)^2 = 225\n\]", "Expand each term: \n\[\n(25 - 20d + 4d^2) + (25 - 10d + d^2) + 25 + (25 + 10d + d^2) + (25 + 20d + 4d^2) = 225\n\]", "Combine like terms: \n- Constants: $ 25×5 = 125 $ → but wait: only five 25s? Wait, first term has 25, others add more — total constants: \n25 (from 25) + 25 (third) + 25 (fourth) + 25 (fifth) = 125? No: actually only the three medium and outer squares contribute variable parts. Let’s properly combine:", "Group constants and $ d $-terms: \n- Constants: $ 25 + 25 + 25 + 25 + 25 = 125 $? No — the 25 is only from the middle term squared. The rest are variable. Correct: \nActually the six fixed 25s? Wait — only five terms. The middle is $ 25 $, and the others expand as: \nSo total sum: \n\[\n(25 - 20d + 4d^2) + (25 - 10d + d^2) + 25 + (25 + 10d + d^2) + (25 + 20d + 4d^2)\n\]", "Now combine: \n- Constants: 25×5 = 125 \n- $ d $ terms: $-20d -10d + 0 + 10d + 20d = 0$ \n- $ d^2 $ terms: $4d^2 + d^2 + 0 + d^2 + 4d^2 = 10d^2$", "So equation becomes: \n\[\n125 + 10d^2 = 225\n\]", "Solve for $ d $: \n\[\n10d^2 = 100 \Rightarrow d^2 = 10 \Rightarrow d = \sqrt{10} \quad (\ ext{positive since population count increases})\n\]", "Now recall $ a = 5 - 2d $. Then: \n\[\na = 5 - 2\sqrt{10}\n\]", "But $ \sqrt{10} \approx 3.16 $, so $ 2\sqrt{10} \approx 6.32 $, hence $ a \approx 5 - 6.32 = -1.32 $ — a negative starting count? Impossible.", "Wait — recheck sign logic. But $ a + 2d = 5 $, $ a = 5 - 2d $. If $ d = \sqrt{10} $, then $ a < 0 $, not valid for bird counts.", "But wait — perhaps $ d $ is negative? Then earlier terms smaller — still problematic. Try $ d < 0 $. Then $ a = 5 - 2d $ could be positive if $ d < 0 $. But $ d^2 = 10 $, so $ d = -\sqrt{10} $? Then $ a = 5 + 2\sqrt{10} \approx 11.32 $, valid.", "But does a negative common difference make sense"]