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Home » Bernhard (title), a noble title derived from the German name Bernhard - Coaching Toolbox

Bernhard (title), a noble title derived from the German name Bernhard - Coaching Toolbox

📅 May 5, 2026 👤 scraface

May 05, 2026

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📌 Multiply by $ 1 - \sqrt{7} $:

📌 Numerator: $ (3 - \sqrt{7})(1 - \sqrt{7}) = 3 - 3\sqrt{7} - \sqrt{7} + 7 = 10 - 4\sqrt{7} $.

📌 Denominator: $ 3(1 - 7) = -18 $.

📌 Ratio: $ \frac{10 - 4\sqrt{7}}{-18} = -\frac{5 - 2\sqrt{7}}{9} $.

📌 Now both are negative — still problem. But ratio of third to first: if $ a + d > a - 3d $, then third > first, but denominator negative implies first negative, so magnitude issue.

📌 But in symmetric AP, if $ d > 0 $, $ a - 3d $ could be negative. To make sense, assume $ a $ large. But both signs give negative — so only one is positive? Try numerically: $ \sqrt{7} \approx 2.6458 $. Then $ a - 3d \approx a - 5.29 $, $ a + 3d \approx a + 7.97 $, $ a + d \approx a + 2.6458 $. For first term positive, $ a > 5.29 $; third term positive. Then $ \frac{a + d}{a - 3d}

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