$$ s=R\left\lbrack\frac{\left(1+i\right)^{n}-1}{i}\right\rbrack;R $$

The formula you've written appears to be related to the calculation of the sum of an annuity. This is typically seen in financial mathematics and is used to determine the future or present value of a series of periodic payments.

### Interpreting the formula:

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

Where:
- $ s $ represents the accumulated value (future value) of the annuity.
- $ R $ is the regular periodic payment (annuity payment).
- $ i $ is the interest rate per period.
- $ n $ is the number of payment periods.

#### Detailed Breakdown:
- $(1 + i)^n$ represents the accumulation factor over $n$ periods.
- The expression $\frac{(1 + i)^n - 1}{i}$ is the future value interest factor for an ordinary annuity (where payments are made at the end of each period).
- Multiplying by $R$ gives the total future value of all the periodic payments, considering the interest accrued.

### Use of the Formula:
This formula is typically used to compute how much a series of regular payments will accumulate to at a future time, assuming a fixed interest rate and fixed payment amounts.

#### Example Calculation:
If you know the values of $R$, $i$, and $n$, you can substitute them into the formula to calculate $s$.

Would you like a specific example or further details on how this formula is derived?

### Related Questions:
1. How is this formula derived from the sum of a geometric series?
2. What changes if the payments are made at the beginning of each period (annuity due)?
3. How does compound interest affect this formula?
4. How do you adjust this formula for continuous compounding?
5. What happens to the future value when the interest rate $i$ approaches 0?

**Tip:** Always ensure that the interest rate $i$ and the number of periods $n$ are expressed in the same units (e.g., monthly payments require a monthly interest rate).

Explore the calculation of the future value of an annuity using the formula s = R[(1 + i)^n - 1]/i. This solution explains the breakdown of periodic payments, interest rate, and time period. Ideal for finance students and professionals.

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