Growth of Bacteria:
📌 Question 1. A certain culture of bacteria initially contains 1 000 bacteria and doubles every hour. How many bacteria are in the culture at the end of 10 hours?
✍ Solution:
Since the number of bacteria doubles every hour and there are initially 1 000, therefore at the end of the first hour there will be 2 000. At the end of the second hour, there will be 4 000 and so on. A table of values will help.

The second row of the table shows a geometric sequence where a₁=2000 and r=2. Using the formula for the nth term of a geometric progression, then,

a_n=a₁⋅r^n-1
a₁₀=2000⋅2^10-1=2000⋅2⁹=2000⋅512=1 024 000

There are 1 024 000 bacteria at the end of 10 hours.

Here notice that we did not start the sequence with 1 000 since it is the initial number of bacteria in the culture and there are no doublings yet.

📌 Example 1. (Biology) A certain bacteria divides every 15 minutes to produce two complete bacteria.
a. If an initial colony contains a population of b₀ bacteria, write an equation that will determine the number of bacteria b_t present after t hours.
b. Suppose a Petri dish contains 12 bacteria. Use the equation found in part (a.) to determine the number of bacteria present 4 hours later.
✍ Solution:
a. Initially, there is 1 bacterium. After 15 minutes, there will be 2 bacteria, after 30 minutes there will be 4 bacteria, after 45 minutes there will be 8 bacteria, and after 1 hour there will be 16 bacteria.
So, in terms of hours, b₀=1 and b₁=16. Find the common ratio.

16÷1=16
Write an explicit formula using r=16.
b_t=b₀ r^t
=b₀⋅16^t
b. Substitute b₀=12 and t=4 into the equation you found in part (a.).
b_t=b₀ r^t
b₄=12⋅16⁴
b₄=786, 432

📌 Example 2: Using a Geometric Series to Model and Solve a Problem
A person takes tablets to cure an ear infection. Each tablet contains 200 mg of an antibiotic. About 12% of the mass of the antibiotic remains in the body when the next tablet is taken. Determine the mass of antibiotic in the body after each number of tablets has been taken. antibiotic.
a) 3 tablets b) 12 tablets
✍ solution:
a) Determine the mass of the antibiotic in the body for 1 to 3 tablets.

The problem can be modelled by a geometric series, which has 3 terms because 3 tablets were taken:
200+200(0.12)+200(0.12)².
The sum is 226.88. So, after taking the 3rd tablet, the total mass of antibiotic in the person’s body is 226.88 mg or just under 227 mg.

b) Determine the sum of a geometric series whose terms are the masses of the antibiotic in the body after 12 tablets. The series is:

200+200(0.12)+200(0.12)²+200(0.12)³+⋯+200(0.12)¹¹.
Use:

Substitute: n=12, t₁=200, r=0.12

The mass of antibiotic in the body after 12 tablets is approximately 227.27 mg, or just over 227 mg.

Fish Population

📌 Question 2. The caretaker of a local fish pond does not allow fishing unless there is an abundance of fish available. He estimates when the fish count passes 1, 000, he will open the pond to the public. The season-opening fish population was estimated at 60 fish, but weekly checks showed the fish count was doubling each week. How many weeks will it take before the pond will open?

✍ One Solution: This is an example of a geometric sequence in which each week the population is multiplied by 2, which means r=2. We need to find when the sum of the fish reaches 1, 000. We can write the formula in explicit form: a_n=60⋅2^n-1. We want to find when a_n=1000.

So, the pond should open in a little over 5 weeks.
Alternate Method:

Alternate Method: y₁=60⋅2^n-1 and y₂=1000, graph and find the x-coordinate of the intersection.