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Home » Substituting $r = \fraca + b - t2$: - jntua results

Substituting $r = \fraca + b - t2$: - jntua results

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 An entomologist is studying a population of beetles in a forest. If the population triples every month starting from 200 beetles, how many beetles are there after 4 months?

📌 Initial population = 200 beetles

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But since the progression is real and the context suggests positive te...
Let $ r = \frac{a + \frac{d}{2}}{a - \frac{3d}{2}} $. Let $ x = \frac{...
r = \frac{1 + \frac{x}{2}}{1 - \frac{3x}{2}}.
Substitute $ x = \sqrt{\frac{28}{9}} = \frac{2\sqrt{7}}{3} $:
r = \frac{1 + \frac{1}{2} \cdot \frac{2\sqrt{7}}{3}}{1 - \frac{3}{2} \...
As before, this evaluates to $ -\frac{5 + 2\sqrt{7}}{9} $, but this is...
Alternatively, let the first term be $ A $, common difference $ d $, s...
A^2 + (A+3d)^2 = A^2 + A^2 + 6Ad + 9d^2 = 2A^2 + 6Ad + 9d^2.
Sum of all terms: $ 4A + 6d $, square: $ 16A^2 + 48Ad + 36d^2 $.
Set equal:

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