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The problem involves compound interest, and we are given the formula for the final amount \( A \), principal \( P \), and interest rate \( i \% \) after 3 years:

\[
A = P \left( 1 + \frac{i}{100} \right)^3
\]

### Part 1: Find \( i \) in terms of \( P \) and \( A \)
We need to express \( i \) in terms of \( A \) and \( P \). Starting from the equation:

\[
A = P \left( 1 + \frac{i}{100} \right)^3
\]

1. Divide both sides by \( P \):

\[
\frac{A}{P} = \left( 1 + \frac{i}{100} \right)^3
\]

2. Take the cube root of both sides:

\[
\left( \frac{A}{P} \right)^{\frac{1}{3}} = 1 + \frac{i}{100}
\]

3. Isolate \( i \):

\[
\frac{i}{100} = \left( \frac{A}{P} \right)^{\frac{1}{3}} - 1
\]

4. Multiply both sides by 100:

\[
i = 100 \left( \left( \frac{A}{P} \right)^{\frac{1}{3}} - 1 \right)
\]

Thus, the interest rate \( i \) in terms of \( A \) and \( P \) is:

\[
i = 100 \left( \left( \frac{A}{P} \right)^{\frac{1}{3}} - 1 \right)
\]

### Part 2: Find \( i \) when \( P = 2500 \) and \( A = 2650 \)

Now, we are given that the principal \( P = 2500 \) and the final amount \( A = 2650 \). Substituting these values into the formula for \( i \):

\[
i = 100 \left( \left( \frac{2650}{2500} \right)^{\frac{1}{3}} - 1 \right)
\]

Let me calculate the value of \( i \).The rate of interest \( i \) is 2.0%, correct to one decimal place.

Would you like further details or have any questions about the solution? 

Here are 5 related questions to expand on this topic:
1. What happens to the interest rate if the investment period is extended from 3 to 5 years?
2. How would the formula change if the interest was compounded monthly rather than annually?
3. What would be the final amount \( A \) if the interest rate increased to 3%?
4. How can you calculate the interest earned on an investment if only the final amount and initial investment are known?
5. What is the difference between simple interest and compound interest?

**Tip:** When dealing with compound interest, small changes in the interest rate or the compounding period can lead to significant differences in the final amount due to the exponential nature of the formula.

The final value of €P, invested for 3 years at i%, is given by the formula A = P(1 + i/100)^3. Find i in terms of P and A. If €2500 invested 3 years ago has a present value of €2650, find the rate of interest, i (correct to one decimal place).

Learn how to find the interest rate i when the final value A of an investment is given after 3 years. This problem uses the compound interest formula A = P(1 + i/100)^3, with an initial investment of €2500 and a final value of €2650.

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