It is observed that the number of ants living in a colony increase by 25% every week. After four weeks of observation, total ants were found to be 6250. How many ants were found when the observation was first made?

Options

A. 1250
B. 2560
C. 3125
D. 3250

Answer

To find the initial number of ants in the colony when the observation was first made, we can follow these steps:

### Step 1: Understand the growth rate
The number of ants increases by 25% every week. This means that if the initial number of ants is \( A \), after one week, the number of ants will be:
\[ A + 0.25A = A \left(1 + \frac{25}{100}\right) = A \left(\frac{5}{4}\right) \]

### Step 2: Determine the number of ants after each week
After one week, the number of ants is:
\[ A_1 = A \left(\frac{5}{4}\right) \]
After two weeks, the number of ants will be:
\[ A_2 = A_1 \left(\frac{5}{4}\right) = A \left(\frac{5}{4}\right) \left(\frac{5}{4}\right) = A \left(\frac{5}{4}\right)^2 \]
After three weeks, the number of ants will be:
\[ A_3 = A_2 \left(\frac{5}{4}\right) = A \left(\frac{5}{4}\right)^2 \left(\frac{5}{4}\right) = A \left(\frac{5}{4}\right)^3 \]
After four weeks, the number of ants will be:
\[ A_4 = A_3 \left(\frac{5}{4}\right) = A \left(\frac{5}{4}\right)^3 \left(\frac{5}{4}\right) = A \left(\frac{5}{4}\right)^4 \]

### Step 3: Set up the equation
According to the problem, after four weeks, the total number of ants is 6250. Therefore, we can set up the equation:
\[ A \left(\frac{5}{4}\right)^4 = 6250 \]

### Step 4: Solve for \( A \)
To find \( A \), we can rearrange the equation:
\[ A = 6250 \div \left(\frac{5}{4}\right)^4 \]
Calculating \( \left(\frac{5}{4}\right)^4 \):
\[ \left(\frac{5}{4}\right)^4 = \frac{5^4}{4^4} = \frac{625}{256} \]
Now substituting this back into the equation for \( A \):
\[ A = 6250 \times \frac{256}{625} \]
The \( 625 \) in the numerator and denominator cancels out:
\[ A = 2560 \]

### Conclusion
The initial number of ants when the observation was first made is:
\[ \boxed{2560} \]