Question

A population of bacteria is doubling every hour. The population size is 500 bacteria.

What is the recursive formula for this situation? Response area

What is the explicit formula for this situation? Response area

Answer 1

Let's address this situation step by step.

### Recursive Formula
The recursive formula expresses the population at time \( t \) in terms of the population at the previous time step.

Let \( P(t) \) represent the population at time \( t \) (in hours). Given that the population doubles every hour, we can express the population as follows:

- Initial population at \( t = 0 \) is \( P(0) = 500 \).
- For each subsequent hour, the population doubles: \( P(t) = 2 \times P(t-1) \).

Thus, the recursive formula is:
\[
P(t) =
\begin{cases}
500 & \text{if } t = 0 \\
2 \times P(t-1) & \text{if } t > 0
\end{cases}
\]

### Explicit Formula
The explicit formula provides a direct calculation for the population at any time \( t \). Given that the population doubles every hour, the population can be described by the formula:

\[
P(t) = P(0) \times 2^t
\]

Substituting the initial population:

\[
P(t) = 500 \times 2^t
\]

### Summary
- **Recursive Formula:**
\[
P(t) =
\begin{cases}
500 & \text{if } t = 0 \\
2 \times P(t-1) & \text{if } t > 0
\end{cases}
\]

- **Explicit Formula:**
\[
P(t) = 500 \times 2^t
\]