Read the following text and answer the following question on the basis of the same :
Growth of population with time shows specific and predictable patters. Two types of growth patter of populaiton are exponetial and logistic growth . When resource in the habitat are unlimited each species has the ability to relize fully its innate potential to grow in number . Then the population gorws in exponential fashion . When the resource are limited growth curve shows an inital slow rate and then it accelerates and finally slows giving the gorwth curve which is sigmoid .
Identify the correct equation that represents the exponential population growth curve:

Options

A. `dN//dt = IN`
B. `dN//dt = rN (K-N//K)`
C. `N_(t) = N_(0) e^(rt)`
D. Both (A) are (C )

Explanation

To identify the correct equation that represents the exponential population growth curve, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Population Growth**: Recognize that population growth can occur in two primary patterns: exponential and logistic. Exponential growth happens under conditions of unlimited resources.

2. **Defining Exponential Growth**: In exponential growth, the rate of change of the population size (dn/dt) is proportional to the current population size (n). This can be expressed mathematically.

3. **Mathematical Representation**: The equation for exponential growth can be represented as:
\[ \frac{dn}{dt} = r \cdot n \]
where:
- \( \frac{dn}{dt} \) is the change in population size over time.
- \( r \) is the intrinsic rate of natural increase (birth rate minus death rate).
- \( n \) is the current population size.

4. **Integrating the Equation**: To find the population size at any time \( t \), we integrate the equation:
\[ \int \frac{1}{n} dn = \int r dt \]
This leads to:
\[ \ln(n) = rt + C \]
where \( C \) is the integration constant.

5. **Solving for Population Size**: Exponentiating both sides gives:
\[ n = e^{rt + C} = e^C \cdot e^{rt} \]
Letting \( e^C \) be the initial population size \( n_0 \), we have:
\[ n = n_0 e^{rt} \]

6. **Final Equation**: Therefore, the equation that represents the exponential growth curve is:
\[ n(t) = n_0 e^{rt} \]

### Conclusion:
The correct equation that represents the exponential population growth curve is \( n(t) = n_0 e^{rt} \).

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