FUTURE VALUE It is now January 1, 2018. Today you will deposit $\mathrm{\$}1,000$ into a savings Account that pays $8\mathrm{\%}$.
a. If the bank compounds interest annually, how much will you have in your account on January 1, 2021?
b. What will your January 1, 2021, balance be if the bank uses quarterly compounding?
c. Suppose you deposit $\mathrm{\$}1,000$ in three payments of $\mathrm{\$}333.333$ each on January 1 of 2019, 2020 and 2021. How much will you have in your account on January 1, 2021, based on? $8\mathrm{\%}$ annual compounding?
d. How much will be in your account if the three payments begin on January 1, 2018?
e. Suppose you deposit three equal payments into your account on January 1 of 2019, 2020 and 2021. Assuming an $8\mathrm{\%}$ interest rate, how large must your payments be to Have the same ending balance as in part a?

Step 1 ：Calculate the future value with annual compounding: FV_a = PV_a * (1 + r_a/n_a)^(n_a*t_a) = 1000 * (1 + 0.08/1)^(1*3) = $1259.71$

Step 2 ：Calculate the future value with quarterly compounding: FV_b = PV_a * (1 + r_a/n_b)^(n_b*t_a) = 1000 * (1 + 0.08/4)^(4*3) = $1268.24$

Step 3 ：Calculate the future value with annual compounding and three equal payments: FV_c = 3 * PV_c * (1 + r_a/n_a)^(n_a*t_a) = 3 * 333.333 * (1 + 0.08/1)^(1*3) = $1082.13$

Step 4 ：Calculate the future value with annual compounding and three equal payments starting from 2018: FV_d = PV_a * (1 + r_a/n_a)^(n_a*t_a) + 2 * PV_c * (1 + r_a/n_a)^(n_a*(t_a-1)) + PV_c * (1 + r_a/n_a)^(n_a*(t_a-2)) = 1000 * (1 + 0.08/1)^(1*3) + 2 * 333.333 * (1 + 0.08/1)^(1*2) + 333.333 * (1 + 0.08/1)^(1*1) = $1168.70$

Step 5 ：Calculate the required payment amount to have the same ending balance as in part a: payment = FV_a / (1 + r_a/n_a)^(n_a*t_a) = 1259.71 / (1 + 0.08/1)^(1*3) = $\boxed{{\displaystyle 473.07}}$