2 An annuity can be modelled by the recurrence relatiorn Vo = 6000, Vn+,-1.005 V

Question

2 An annuity can be modelled by the recurrence relatiorn Vo = 6000, Vn+,-1.005 V,,-15 18 where V, is the balance of the annuity after n payments have been made. The annuity is to be fully repaid after four payments. a Use your calculator to determine recursively the balance of the annuity after the two payments have been made. Give your answer to the nearest cent b Is the annuity fully paid out after four payments of $1518 have been received? If not, how much will the last payment have to be to ensure that the balance of the annuity is zero after five payments?

Explanation / Answer

2. a. Vn is the balance of the annuity after the payment is made. So the initial value of the principle amount is 6000

Again, 1518 is the amount which will be deducted from the principle amount so,

V(n+1) = 1.005*6000 – 1518 = 4512

So the amount of annuity for the next 2 payments will be,

V (n+2) = 1.005*4512 – 1518 = 3016

V (n+3) = 1.005*3016 – 1518 = 1513

So value of annuity for the next 2 payments will be (3016 & 1513)

2. b. Yes in the 4th payment (i.e.) {V (n+4) = 1513*1.005 – 1518 = 3} the annuity is almost paid out fully only balance remaining is 3 which is obtained because of the approximation of the figure. So if the company wants to carry the approximation in the 5th payment too then it will have to pay just a value of 3, otherwise it will be cleared in the 4th payment only making the annuity value “Zero”.

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