Modelling and analysing reducing-balance loads and annuities - REVISION

The recurrence relation that generates the sequence 5, 9, 17, 33, 65, … is

• V₀ = 5, V_n+1 = 5V_n

• V₀ = 5, V_n+1 = 14 – V_n

• V₀ = 5, V_n+1 = V_n + 4

• V₀ = 5, V_n+1 = 2V_n – 1

• V₀ = 5, V_n+1 = 3V_n – 6

The recurrence relation V₀ = 10 000, V_n+1 = 1.012V_n – 2000 models a reducing-balance loan with payments of $2000 each quarter, where V_n is the value of the loan after n quarters. The annual interest rate for this loan is

• 1.2%

• 1.4%

• 3.12%

• 4.8%

• 14.4%

The recurrence relation V₀ = 375 000, V_n+1 = 0.918V_n is used to calculate the value of farm machinery after n years, V_n , that is depreciating in value using a reducing-balance method. The value of the machinery after eight years, V₈, correct to the nearest dollar, is

• 159 389

• 173 626

• 189 135

• 206 030

• 224 433

A car purchased for $25 000 will be depreciated at the rate of 8.2% per annum, using a flat-rate method. The car will be sold when it’s value first drops below $10 000. The car will be sold after

• 8 years

• 9 years

• 10 years

• 11 years

• 12 years

The number of years it will take an investment of $5000, earning compound interest at the rate of 6.2% per annum to grow above $7000 is

• 5

• 6

• 7

• 8

• 9

A reducing-balance loan of $6000 is charged compound interest at the rate of 7.68% per annum, compounding monthly. The loan will be repaid with monthly payments of $400. A recurrence relation that models the value of the loan after n months, V_n, is

• V₀ = 6000, V_n+1 = 0.9232V_n + 400

• V₀= 6000, V_n+1 = 0.914V_n – 400

• V₀= 6000, V_n+1 = 0.9936V_n + 400

• V₀= 6000, V_n+1 = 1.086V_n – 400

• V₀= 6000, V_n+1 = 1.0064V_n – 400

The value of a loan charged compound interest every year is modelled using the recurrence relation V₀ = 15 000, V_n+1 = 1.08V_n, where V_n is the value of the loan after n years. A rule for the value of the loan after n years is

• V_n = 15 000 + 1.08n

• V_n = 15 000 + 1200n

• V_n = (1.08)ⁿ x 2000

• V_n = (1.08)ⁿ x 15 000

• V_n = (15 000)ⁿ x 1.08

A perpetuity investment is established that will provide a monthly payment of $400. If the investment is paid compounding interest at the rate of 6.1% per annum, compounding monthly, the principal that must be invested is closest to

• $6500

• $26 230

• $65 570

• $78 700

• $170 400

Brian invested $50 000 into an account that earns compound interest at the rate of 4.6% per annum, compounding fortnightly. After one year, the value of the investment has grown to $58 997. The amount Brian is adding to his investment every fortnight is closest to

• $250

• $270

• $540

• $650

• $890

The value of V₃ in the sequence defined by the recurrence relation V₀ = 5, V_n+1 = V_n + 6 is:

• 5

• 11

• 17

• 23

• 29