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:

$$\begin{gathered} q=\frac{11}{10}=1.1 \\ {q}^{n-1}=3 \\ (n-1)\cdot \ln q=\ln 3 \\ {n}_1=\ln 3/\ln q+1=\ln 3/\ln 1.1+1\doteq 12.5267 \\ n=\lceil n_1\rceil =\lceil 12.5267\rceil =13 \\ \text{ Verifying Solution: } \\ {a}_1=1 \\ {a}_2=a_1\cdot q=1\cdot \frac{11}{10}=\frac{1\cdot 11}{10}=\frac{11}{10}=1.1 \\ {a}_3=a_2\cdot q=\frac{11}{10}\cdot \frac{11}{10}=\frac{11\cdot 11}{10\cdot 10}=\frac{121}{100}=1.21 \\ {a}_4=a_3\cdot q=\frac{121}{100}\cdot \frac{11}{10}=\frac{121\cdot 11}{100\cdot 10}=\frac{1331}{1000}=1.331 \\ {a}_5=a_4\cdot q=1.331\cdot 1.1=1.4641 \\ {a}_6=a_5\cdot q=1.4641\cdot 1.1\doteq 1.6105 \\ {a}_7=a_6\cdot q=1.6105\cdot 1.1\doteq 1.7716 \\ {a}_8=a_7\cdot q=1.7716\cdot 1.1\doteq 1.9487 \\ {a}_9=a_8\cdot q=1.9487\cdot 1.1\doteq 2.1436 \\ {a}_{10}=a_9\cdot q=2.1436\cdot 1.1\doteq 2.3579 \\ {a}_{11}=a_{10}\cdot q=2.3579\cdot 1.1\doteq 2.5937 \\ {a}_{12}=a_{11}\cdot q=2.5937\cdot 1.1\doteq 2.8531 \\ {a}_{13}=a_{12}\cdot q=2.8531\cdot 1.1\doteq 3.1384 \end{gathered}$$

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