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The equation you provided is:

\[
F = A \left[ \frac{(1 + i)^n - 1}{i} \right]
\]

You want to solve this equation for \(i\). Let's go step by step.

### 1. Multiply both sides of the equation by \(i\) to eliminate the denominator:

\[
F \cdot i = A \left[ (1 + i)^n - 1 \right]
\]

### 2. Divide both sides by \(A\):

\[
\frac{F \cdot i}{A} = (1 + i)^n - 1
\]

### 3. Add \(1\) to both sides:

\[
\frac{F \cdot i}{A} + 1 = (1 + i)^n
\]

### 4. Take the \(n\)-th root of both sides:

\[
\left( \frac{F \cdot i}{A} + 1 \right)^{\frac{1}{n}} = 1 + i
\]

### 5. Subtract \(1\) from both sides:

\[
\left( \frac{F \cdot i}{A} + 1 \right)^{\frac{1}{n}} - 1 = i
\]

At this point, \(i\) is isolated, but we need to simplify the expression involving \(F\), \(A\), and \(n\) depending on the values or given context.

Would you like a further breakdown or clarification of this process? 

Here are five follow-up questions to deepen understanding:
1. How is the interest rate \(i\) typically used in financial mathematics?
2. What are common applications of this formula in finance (e.g., annuities)?
3. How does the exponent \(n\) influence the result?
4. What would happen if \(F = A\)? How does that change the solution for \(i\)?
5. Can you solve for \(i\) if specific values of \(F\), \(A\), and \(n\) are provided?

**Tip:** In financial mathematics, the term \( (1+i)^n \) often represents compound growth, with \(i\) as the interest rate and \(n\) as the number of periods.

Learn how to solve for the interest rate (i) in the compound interest formula F = A[(1+i)^n - 1]/i with a step-by-step solution. This process involves algebraic manipulation, exponent properties, and financial mathematics. Suitable for advanced high school or college students.

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