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The problem asks to find the number of periods $n$ using the future value formula for a series of payments (annuities). The given values are:

- Future value $FV = 8000$
- Interest rate $i = 0.04$
- Payment per period $PMT = 400$
- $n = ?$

### Future Value Formula for Annuity:

The formula to calculate the future value of an ordinary annuity is:

\[
FV = PMT \times \frac{(1 + i)^n - 1}{i}
\]

Where:
- $FV$ is the future value
- $PMT$ is the payment per period
- $i$ is the interest rate per period
- $n$ is the number of periods

We need to solve for $n$. Rearranging the formula:

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

\[
(1 + i)^n = 1 + \frac{FV \cdot i}{PMT}
\]

\[
n = \frac{\log \left(1 + \frac{FV \cdot i}{PMT}\right)}{\log(1 + i)}
\]

### Substituting the given values:
- $FV = 8000$
- $i = 0.04$
- $PMT = 400$

We can now calculate $n$. Let me do that calculation.The number of periods $n$ is approximately 15 when rounded up to the nearest integer.

Would you like more details on the steps or any further clarification?

Here are 5 related questions to further explore this topic:
1. How would the future value change if the payment per period increased?
2. What is the effect of changing the interest rate on the number of periods required to reach the future value?
3. Can the same formula be used for compound interest? If not, what would change?
4. What happens if the payments are made at the beginning of each period instead of the end?
5. How would the formula adjust for different compounding frequencies (e.g., monthly, quarterly)?

**Tip:** When solving for the number of periods $n$, always make sure the interest rate is consistent with the payment period (e.g., yearly payments with yearly interest rate).

Use the future value formula to find the indicated value. FV = 8,000; i = 0.04; PMT = $400; n = ?

Learn how to determine the number of periods (n) using the future value formula for an ordinary annuity. In this problem, we calculate n for FV = 8,000, an interest rate of 0.04, and a payment per period of $400.

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