To find the population at the beginning of the first year, we can use the formula for compound interest, which is applicable to population growth:

$P = \frac{A}{(1 + r)^n}$

Where:

• $P$ is the initial population (what we are solving for).
• $A$ is the population at the end of $n$ years.
• $r$ is the annual growth rate (expressed as a decimal).
• $n$ is the number of years.

Given:

• $A = 10,000$ (population at the end of 3 years),
• $r = 25\% = 0.25$,
• $n = 3$ years.

Now, substitute the values into the formula:

$P = \frac{10,000}{(1 + 0.25)^3}$

Calculate the denominator:

$(1 + 0.25)^3 = (1.25)^3 = 1.953125$

Thus:

$P = \frac{10,000}{1.953125} \approx 5,119.87$

So, the population at the beginning of the first year was approximately 5,120 people.

Would you like any further details or have any questions?

Here are some related questions to consider:

1. What would the population be at the end of 4 years with the same growth rate?
2. How does the population change if the growth rate drops to 20% annually?
3. What is the formula for continuous population growth?
4. How would the population change if instead of increasing, it decreased by 10% annually?
5. How long would it take for the population to double with a 25% annual growth rate?
6. What is the effect of compound interest in financial contexts compared to population growth?
7. How can you model a population that grows at different rates in different years?
8. What are some real-life factors that might cause a population to grow at a constant annual rate?

Tip: When dealing with percentage increases or decreases, always convert the percentage to a decimal before performing calculations.