Ann and Tom want to establish a fund for their grandson's college education. What lump sum must they deposit at a $10 \%$ annual interest rate, compounded annually, in order to have $\$ 20,000$ in the fund at the end of 10 years? They should deposit $\$ \square$. (Round up to the nearest cent.)

Step 1 ：Ann and Tom want to establish a fund for their grandson's college education. They want to know what lump sum they must deposit at a $10 \%$ annual interest rate, compounded annually, in order to have $\$ 20,000$ in the fund at the end of 10 years.

Step 2 ：The question is asking for the present value of a future sum of money. The formula for calculating the present value is: \(PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}\) where: \(PV\) is the present value, \(FV\) is the future value, \(r\) is the annual interest rate (in decimal), \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested for in years.

Step 3 ：In this case, \(FV = 20000\), \(r = 0.1\), \(n = 1\) (since interest is compounded annually), and \(t = 10\) years.

Step 4 ：We can plug these values into the formula to find the present value, which is the amount that Ann and Tom must deposit now: \(PV = \frac{20000}{(1 + \frac{0.1}{1})^{1*10}}\)

Step 5 ：Calculating the above expression, we find that \(PV = 7710.865788590628\)

Step 6 ：Rounding up to the nearest cent, we get \(PV = 7710.87\)

Step 7 ：Final Answer: They should deposit \(\boxed{\$7710.87}\)