Project 1.4. Compound Interest When you save money in a bank account, interest is compounded periodically, which means that periodically the interest you earn is added to your account so that you start earning interest on your interest. 1. Suppose you deposit $1, 000 into a bank account that has a 5% annual interest rate. Further suppose that interest is compounded monthly. Thus initially you have $1,000; at the end of one month your account has $1,000 + 0.05/12 x 1,000 or $1,004.17; at the end of two months your account has $1.004.17 + 0.05/12 1,004.17 or $1,008.35; and so on. Notice that even though the model's scale is in years, the time step is one month. Notice further that since the interest rate is given in years (the time scale), we need to divide by 12 to compute the interest rate per time step. Set up a recurrence relation for the amount of money a(t + 1), given that we know the amount of money in the account after t time steps 2(t). Track the amount of money in the account over two years. 2. Find a closed form solution for 2(), the amount in the account after i time steps, using data and assumptions from part 1. 3. Find the recurrence relation for the amount of money in the account a if P dollars are invested at an annual interest rate r which is compounded n times per year for i time steps. 4. Solve to find a closed-form solution for the recurrence relation in part 3. Express the function in terms of t years, instead of i time steps. 5. Suppose now that you deposit $1, 000 in an account that is compounded monthly but which has a variable interest rate. Track your account over two years using the monthly interest rates given in Table 1.2. The rate at t= 0 corresponds to the rate from the moment of deposit until the first compounding period.