Question

1. fill in the table below using the context of the problem, find the common difference/ratio, and then write an explicit equation that represents the problem for any amount of 12 - hour periods, n.

a scientist is growing bacteria in a lab. the amount of bacteria increases by 50% every 12 hour period. the scientist starts with 16 bacteria.

# of 12 - hour periods	1	2	3	4
# of bacteria	16

what is the explicit equation that could be used to model the amount of bacteria, f(n), after n 12 - hour periods?
explicit equation:

Explanation:

Step1: Find the common ratio

Since the amount of bacteria increases by 50% (or 0.5) every 12 - hour period, the common ratio $r=1 + 0.5=1.5$.

Step2: Calculate the number of bacteria for each period

For $n = 1$, the number of bacteria $a_1=16$.
For $n = 2$, $a_2=a_1\times r=16\times1.5 = 24$.
For $n = 3$, $a_3=a_2\times r=24\times1.5 = 36$.
For $n = 4$, $a_4=a_3\times r=36\times1.5 = 54$.

Step3: Write the explicit formula

The general form of a geometric - sequence explicit formula is $a_n=a_1\times r^{n - 1}$. Here, $a_1 = 16$ and $r = 1.5$, so the explicit equation for the amount of bacteria $f(n)$ after $n$ 12 - hour periods is $f(n)=16\times(1.5)^{n - 1}$.

Answer:

# of 12 - hour periods	1	2	3	4

Explicit Equation: $f(n)=16\times(1.5)^{n - 1}$