Microorganisms

The first generation of microorganisms has a population of 13500 members. Each subsequent generation is 11/10 times the previous one. Find out how many generations will reach at least three times members of the first generation.

Correct answer:

n = 13

Step-by-step explanation:

q = \frac{1 1}{1 0} = 1.1 q^{n - 1} = 3 (n - 1) \cdot ln q = ln 3 n_{1} = ln 3 / ln q + 1 = ln 3 / ln 1.1 + 1 ≐ 12.5267 n = ⌈ n_{1} ⌉ = ⌈ 12.5267 ⌉ = 13 Verifying Solution: a_{1} = 1 a_{2} = a_{1} \cdot q = 1 \cdot \frac{1 1}{1 0} = \frac{1 \cdot 1 1}{1 0} = \frac{1 1}{1 0} = 1.1 a_{3} = a_{2} \cdot q = \frac{1 1}{1 0} \cdot \frac{1 1}{1 0} = \frac{1 1 \cdot 1 1}{1 0 \cdot 1 0} = \frac{1 2 1}{1 0 0} = 1.21 a_{4} = a_{3} \cdot q = \frac{1 2 1}{1 0 0} \cdot \frac{1 1}{1 0} = \frac{1 2 1 \cdot 1 1}{1 0 0 \cdot 1 0} = \frac{1 3 3 1}{1 0 0 0} = 1.331 a_{5} = a_{4} \cdot q = 1.331 \cdot 1.1 = 1.4641 a_{6} = a_{5} \cdot q = 1.4641 \cdot 1.1 ≐ 1.6105 a_{7} = a_{6} \cdot q = 1.6105 \cdot 1.1 ≐ 1.7716 a_{8} = a_{7} \cdot q = 1.7716 \cdot 1.1 ≐ 1.9487 a_{9} = a_{8} \cdot q = 1.9487 \cdot 1.1 ≐ 2.1436 a_{10} = a_{9} \cdot q = 2.1436 \cdot 1.1 ≐ 2.3579 a_{11} = a_{10} \cdot q = 2.3579 \cdot 1.1 ≐ 2.5937 a_{12} = a_{11} \cdot q = 2.5937 \cdot 1.1 ≐ 2.8531 a_{13} = a_{12} \cdot q = 2.8531 \cdot 1.1 ≐ 3.1384

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