Please explain it in details, thanks.

In this exercise we will derive the formulas for the present value and future value of an ordinary annuity. Suppose there are N pe- riods and that today, time t = 0, is when the ordinary annuity is contracted. This means that a set of equal payments A are to be paid or received at times t = 1, 2, . . ., N. The present value is computed today at t = 0, one period before the first payment; the future value is computed at time t = N , immediately upon the last payment of the annuity.1 (a) First establish the formula for the partial sum of a geometric series: N SN 2 E ark (1-1) k=0 =a+ar+ar2+---+arN (1.2) _ NH 2 a (11%) , r a 1. (1.3) This relies on a simple trick: multiply equation (1.2) by the com- mon factor 1' and subtract it from (1.2) to obtain an expression for SN rSN. Then simply solve for SN. (b) As we saw in class, the future value (value at time t = N) of an ordinary annuity can be found by considering the future values of each of the N payments: the rst payment A received at time t = 1 compounds for N 1 periods at rate r; the second payment A received at time t = 2 compounds for N 2 periods at rate r; and so on, until the last payment A at t = N , which does not compoundit's value at time t = N is simply A. Write the future value at time t = N of an ordinary annuity as a partial sum of a geometric series and use the formula from (a) to find a closed form expression for it. (c) Similarly, we can compute the present value (value at time t = 0) of an ordinary annuity by considering the present values of each of the N payments. Write the present value of an ordinary annuity as a partial sum of a geometric series and use the formula from (a) to find a closed form expression for it