RECURSION & FINANCIAL MODELLING SUMMARY Recurrence Relation Rule Notes LINEAR GROWTH / DECAY A. Simple Interest Loans / Investments Vo = principal AND Vn+1 = Vn + D, D= r n = number of years 100 X Vo Vn = Vo + nD where D = 100 ~ X Vo r = interest rate per year D = amount of interest ($) Vn = balance of loan/investment after n years B. Flat Rate Depreciation Vo = initial value of the asset AND Vn+1 = Vn - D Vn = Vo - nD where D = r Vn = value of the asset after n years 100 X Vo where D = X Vo 100 C. Unit Cost Depreciation Vo = initial value of the asset AND Vn+1 = Vn - D Vn = Vo - nD D = cost per unit of use GEOMETRIC GROWTH / DECAY A. Compound Interest Loans / Investments n = number of compounding periods Vo = principal AND Vn+1 = RV, where R = 1+ r Vn = R" Vo where R = 1+ 100 100 r = interest rate per compounding period Vo = principal Vn = value of loan / investment after n compounding periods B. Reducing Balance Depreciation Vo = initial value AND Vn+1 = R Vn , where R = 1- Vn = R" Vo where R = 1- r N = number of years 100 100 Vn = value of asset after n years Vo = principal COMBINED LINEAR & GEOMETRIC GROWTH / DECAY Reducing Balance Loan Vo = principal AND Vn+1 = RVn - D D = regular payment per payment periord where R = 1+ r Rule for this is not required Vn = balance of the loan after n payments r = interest rate per compounding period Interest-only loan (reducing balance loan where interest = payment)Vo = principal and Vn+1 = Vn r D = regular payment per payment period Vo = principal D = 100 -xVo r = interest rate per compounding period Annuity (money is invested and gradually withdrawn over a period of time) Vo = principal and Vn+1 = R Vn - D D = regular payment received per payment period r r = interest rate per compounding period where R = 1 + 100 Perpetuity (annuity where interest = payment received) Vo = principal AND Vn+1 = Vn D = regular payment received per payment period Vo = principal D = - r 100 - xVo r = interest rate per compounding period Compound Interest Investments with Regular Additions to the Principal (Annuity Investment) Vo = principal and Vn+1 = R Vn + D D = regular addition to the principal r = interest rate per compounding period where R = 1+ 100DECODE VCE FURTHER MATHEMATICS 48 Question 7 Lezly has won $100 000 000 at the pokies. He decides to invest this money in an annuity in order to provide a better life for his child. The annuity will earn interest at a rate of 3.7% per annum compounding fortnightly, and will last for exactly 55 years. The annuity will pay $N fortnighty. The value of N is closest to A. 163 736 N= 55 x 26 B. 193 532 1: 3.7 C. 444 853 PV : - 1000000 00 D. 1891 557 PMT =9 16375- 73 E. 2248 387 FV = O Question 8 Ply - 26 The sequence Sn can be defined by the formula Sn = ant + b for all n 2 0, where a and b are constants. Which of the following is not a possible recurrence relation that Sn could satisfy? A. Sn+1 = 25n B. Sn+1 = Sn C. Sn+1 = 0Sn D. Sn+1 = Sn XSn E. Sn+1 = Sn + 1 END OF TEST Download the solutions manual at www.decodeguides.com.au or scan the QR code below! Volume 3 Solutions Manual10 00 46 DECODE VCE FURTHER MATHEMATICS Instructions Answer all questions by circling one of the bolded letters in the given list of responses. Choose the response that is correct for the question. A correct answer scores 1; an incorrect answer scores 0. Marks will not be deducted for incorrect answers. No marks will be given if more than one answer is completed for any question. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Question 1 Consider the recurrence relation below. Co = 3.3, Cn+1 = 3.3Cn+ 3 The value of C4 is closest to A. 48.83 B. 48.84 C. 164.16 D. 544.73 E. 544.74 Question 2 If the sequence Sn satisfies the recurrence relation Sn+1 = 25, + 1, then S3 is equal to A. 250 + 7 B. 450 + 2 C. 850 + 7 D. 450 + 3 E. 850 +5 Question 3 Suppose that Vn is the value of a perpetuity after n payments, and that Vn satisfies the recurrence relation Vn+1 = 1.008Vn - 400. The value of Vo is A. 40 000 B. 50 000 400 = C. 60 000 D. 70 000 E. 80 000 910 Volume 3 Solutions Manual Did Scan QR codeTM to open