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Home » The sum of the first n natural numbers is 210. Find n. - Abbey Badges

The sum of the first n natural numbers is 210. Find n. - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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Let $ A = 4x $, $ B = 2x $:
\cos 4x - \cos 2x = -2 \sin(3x)\sin(x)
f(x) = 1 + rac{1}{2}(-2 \sin 3x \sin x) = 1 - \sin 3x \sin x
We now seek the minimum value of $ f(x) = 1 - \sin 3x \sin x $. Since...
- 1 = 0
We verify this value is attainable. Suppose $ \sin 3x = 1 $ and $ \sin...
ight) = -1
e 1 $. Not simultaneous. Try to find $ x $ such that $ \sin 3x \sin x...
ight) = -1 $. Then $ \sin 3x \sin x = -1 $.
Try $ \sin x = 1 $, $ \sin 3x = 1 $: requires $ x = rac{\pi}{2} $, bu...

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