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

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

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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+ \sin 2x + rac{4}{\sin^2 2x}
Let \(t = \sin 2x\), \(0 < t \leq 1\) (since \(x \in (0, \pi/2)\Righta...
Define \(f(t) = 1 + t + rac{4}{t^2}\), \(0 < t \leq 1\)
Derivative: \(f'(t) = 1 - rac{8}{t^3}\)
Set to zero: \(1 = rac{8}{t^3} \Rightarrow t^3 = 8 \Rightarrow t = 2\...
Then \(f(1) = 1 + 1 + 4 = 6\)
As \(t o 0^+\), \(f(t) o \infty\), but wait â is \(t\) bounded bel...
But in domain \(0 < x < \pi/2\), \(\sin 2x = 0\) is not attained, but...
But question says "find the maximum", implying it exists. Contradictio...
Wait â at \(x = \pi/4\), \(\sin 2x = 1\), \(f = 1 + 1 + 4 = 6\), val...

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