Question

The population size of a culture of bacteria triples every hour such that its size is modeled by the sequence a Subscript nequals100 left parenthesis 3 right parenthesis Superscript n minus 1​, where n is the number of the hour just beginning. Find the size of the culture at the beginning of the fourth hour and the size of the culture at the beginning of the first hour.
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Part 1
At the begining of the fourth​ hour, the size of the colony is
  
enter your response here bacteria.

Answer 1

To find the size of the bacteria culture at the beginning of the fourth hour and the first hour, we need to evaluate the expression given in the formula:

\[
a_n = 100 \cdot (3)^{n-1}
\]

### Part 1: Size at the beginning of the fourth hour (n=4)
At the beginning of the fourth hour, use \(n = 4\):

\[
a_4 = 100 \cdot (3)^{4-1} = 100 \cdot (3)^{3}
\]

Now calculate \(3^3\):

\[
3^3 = 27
\]

So,

\[
a_4 = 100 \cdot 27 = 2700
\]

Thus, the size of the colony at the beginning of the fourth hour is **2700 bacteria**.

### Part 2: Size at the beginning of the first hour (n=1)
At the beginning of the first hour, use \(n = 1\):

\[
a_1 = 100 \cdot (3)^{1-1} = 100 \cdot (3)^{0}
\]

Since \(3^0 = 1\):

\[
a_1 = 100 \cdot 1 = 100
\]

Thus, the size of the colony at the beginning of the first hour is **100 bacteria**.