1. **State the problem:** We need to find the present value $P$ that will grow to a future value $A = 29000$ with an interest rate of 2% per quarter compounded quarterly for 13 quarters.
2. **Formula used:** The formula for compound interest is $$A = P \left(1 + \frac{r}{n}\right)^{nt}$$ where:
- $A$ is the amount of money accumulated after $n t$ periods,
- $P$ is the principal (present value),
- $r$ is the annual interest rate (decimal),
- $n$ is the number of times interest is compounded per year,
- $t$ is the time in years.
Since the interest is compounded quarterly and the rate is given per quarter, we can simplify to $$A = P (1 + r)^t$$ where $r = 0.02$ and $t = 13$ quarters.
3. **Rearrange to solve for $P$:**
$$P = \frac{A}{(1 + r)^t}$$
4. **Substitute values:**
$$P = \frac{29000}{(1 + 0.02)^{13}} = \frac{29000}{(1.02)^{13}}$$
5. **Calculate the denominator:**
$$(1.02)^{13} \approx 1.02^{13} = 1.29687$$
6. **Calculate $P$:**
$$P = \frac{29000}{1.29687} \approx 22356.68$$
7. **Final answer:** The present value is approximately **22356.68** rounded to the nearest cent.
Present Value 47A1A6
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