A grandparent puts $5,000 into a college education fund for a grandchild. If the fund earns 3.75% annual interest compounded daily, what is the value (in dollars) of the account after 16 years? Assume all years have 365 days.

To calculate the future value of an investment with compound interest, we can use the formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

where:
- $A$ is the amount of money accumulated after n years, including interest.
- $P$ is the principal amount (the initial amount of money).
- $r$ is the annual interest rate (decimal).
- $n$ is the number of times that interest is compounded per year.
- $t$ is the number of years the money is invested or borrowed.

Given:
- $P = 5000$
- $r = 0.0375$ (3.75% as a decimal)
- $n = 365$ (compounded daily)
- $t = 16$

Plugging in the values:

\[
A = 5000 \left(1 + \frac{0.0375}{365}\right)^{365 \times 16}
\]

Now we calculate:

1. Calculate $\frac{r}{n}$:
   \[
   \frac{0.0375}{365} \approx 0.00010274
   \]

2. Calculate $nt$:
   \[
   365 \times 16 = 5840
   \]

3. Substitute back into the formula:
   \[
   A = 5000 \left(1 + 0.00010274\right)^{5840}
   \]
   \[
   A = 5000 \left(1.00010274\right)^{5840}
   \]

4. Now calculate $\left(1.00010274\right)^{5840}$:
   \[
   \left(1.00010274\right)^{5840} \approx 1.776618
   \]

5. Finally, calculate $A$:
   \[
   A \approx 5000 \times 1.776618 \approx 8883.09
   \]

Thus, the value of the account after 16 years is approximately **$8,883.09**.

If you have any questions or need further details, let me know! Here are some related questions you might consider:

1. How would the account value change if the interest rate were 4% instead of 3.75%?
2. What would be the total interest earned over the 16 years?
3. How does the compounding frequency (daily vs. annually) affect the final amount?
4. If an additional $1,000 is deposited each year, how would that change the final value?
5. What is the formula for calculating simple interest, and how does it differ from compound interest? 

**Tip:** Always compare different interest rates and compounding frequencies to maximize your investment growth.

Discover how a $5,000 investment grows to approximately $8,883.09 after 16 years with a 3.75% daily compounded interest rate. This detailed solution utilizes the compound interest formula and explores financial mathematics concepts.

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