If an amount P is deposited today into an account earning interest at a rate of i compounded annually, then the balance in the account will grow by a factor of 1 + i every year.

a) Suppose that x is deposited into an account earning interest at the rate of i compounded annually. Find an expression for the balance in the account t years later.

b) Assuming an interest rate of i compounded annually, find an expression for the amount of money that would have to be deposited now in order to grow to an amount x at a time t years from now. This is called the present value of that payment.

c) An annuity is a regular series of payments, and the present value of the annuity is equal to the sum of present values of the payments. Suppose that an annuity pays $10,000 at the end of every year for 20 years with an interest rate of i = 0.05. Express the present value of the annuity as a finite geometric series and find its sum.

d) A perpetuity is a regular series of payments that continues indefinitely. Again assuming 5% interest (that is i = 0.05), find the present value of a perpetuity paying $10,000 at the end of each year forever.