In a certain experiment the count of bacteria was increasing at the rate of `2.5%` per hour. Initially, the count was `5,12,000`. Find the bacteria at the end of 2 hours.

Answer

To solve the problem of finding the count of bacteria after 2 hours given an initial count and a percentage increase, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Initial count of bacteria (P) = 5,12,000
- Rate of increase (R) = 2.5% per hour
- Time (n) = 2 hours

2. **Convert the Percentage to Decimal**:
- To use the percentage in calculations, convert it to decimal form:
\[ R = \frac{2.5}{100} = 0.025 \]

3. **Use the Compound Interest Formula**:
- The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{n}\right)^{nt} \]
- In our case, since the rate is per hour and we are calculating for 2 hours, we can simplify it to:
\[ A = P \left(1 + R\right)^t \]
- Here, \(t\) is the number of hours (2 hours).

4. **Substitute the Values into the Formula**:
- Substitute \(P\), \(R\), and \(t\) into the formula:
\[ A = 5,12,000 \left(1 + 0.025\right)^2 \]

5. **Calculate the Value Inside the Parentheses**:
- Calculate \(1 + 0.025\):
\[ 1 + 0.025 = 1.025 \]

6. **Raise the Result to the Power of 2**:
- Now calculate \(1.025^2\):
\[ 1.025^2 = 1.050625 \]

7. **Multiply by the Initial Count**:
- Now, multiply this result by the initial count:
\[ A = 5,12,000 \times 1.050625 \]

8. **Perform the Multiplication**:
- Calculate the final amount:
\[ A = 5,37,920 \]

9. **Conclusion**:
- The count of bacteria at the end of 2 hours is **5,37,920**.