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In a research lab, an antibiotic is tested by exposing it to a bacteria. Because of the presence of the antibiotic, the number of bacteria in each new generation is \(70\%\) of the number in the previous generation. There were \(24\thinspace 000\) bacteria initially.

1. Find the number of bacteria in the first five generations.
   
   
   Solution

   The first five terms are:
   \(\begin{align}
    t_1 &= 24\thinspace 000 \\
    t_2 &= 24\thinspace 000\left( {0.70} \right) = 16\thinspace 800 \\
    t_3 &= 16\thinspace 800\left( {0.70} \right) = 11\thinspace 760 \\
    t_4 &= 11\thinspace 760\left( {0.70} \right) = 8\thinspace232 \\
    t_5 &= 8\thinspace 232\left( {0.70} \right) = 5\thinspace762 \\
    \end{align} \)
   
   
2. Assuming nothing inhibits the growth of bacteria, what is the total number of bacteria in the first five generations?
   
   
   Solution

   While you could add up the terms from above, a more efficient method is to use a sum of geometric series formula.
   
   \(\begin{align}
    S_n &= \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}} \\
    S_5 &= \frac{{24\thinspace000\left[ {\left( {0.70} \right)^5 - 1} \right]}}{{0.70 - 1}} \\
    S_5 &= 66\thinspace554.4 \\
    \end{align}\)
   
   The total population in the first five generations is \(66\thinspace 554\).
   
   
   
3. Assuming nothing inhibits the growth of bacteria, has the population reached \(100\thinspace 000\) by the tenth generation?
   
   
   Solution

   \(\begin{align}
    S_n &= \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}} \\
    S_{10} &= \frac{{24\thinspace000\left[ {\left( {0.70} \right)^{10} - 1} \right]}}{{0.70 - 1}} \\
    S_{10} &= 77 \thinspace740.19081 \\
    S_{10} &\buildrel\textstyle.\over =  77\thinspace 740 \\
    \end{align}\)
   
   No, the population has only reached \(77\thinspace 740\).
   
   
   
4. If this growth trend continues indefinitely, how many bacteria will there be in the population?
   
   
   Solution

   Because \(-1 < r < 1 \), you can use the sum of an infinite geometric series formula.
   
   \(\begin{align}
    S_\infty &= \frac{{t_1 }}{{1 - r}} \\
    S_\infty &= \frac{{24\thinspace 000}}{{1 - 0.70}} \\
    S_\infty &= 80\thinspace 000 \\ 
    \end{align}\)
   
   If the trend continues indefinitely, there will be \(80\thinspace 000\) bacteria.
   

For more examples on geometric series, please read through pp. 58 - 63 in Pre-Calculus 11.