2024 Recursion VCAA NHT Exam 1

Mel bought a new car for $60,000. She will depreciate the value of the car using the reducing balance method. A recurrence relation that models the year-to-year value of her car, M_n, is

M₀ = 60,000, M_n+1 = 0.85 M_n. An equivalent rule to determine the value of the car after n years is

• A. M_n = 60,000 - 0.85n

• B.M_n = 60,000 + 0.85n

• C. M_n = 60,000 + 0.85ⁿ

• D. M_n = 60,000 × 0.85^n-1

• E. M_n = 60,000 × 0.85ⁿ

Answer explanation

The correct choice is E because the reducing balance method involves multiplying the previous value by a constant factor, which is 0.85 in this case, for each year of depreciation.

A sequence of numbers is generated by a recurrence relation of the form T₀ = 5, T_n+1 = k - T_n.

All terms of the sequence have the same value. The constant k is equal to

• A. -10

• B. -5

• C. 0

• D. 5

• E. 10

Answer explanation

Since all terms of the sequence have the same value, the value must be a fixed point of the recurrence relation. Substituting T_n = T_n+1 into the relation gives T = k - T, which simplifies to 2T = k. Since T = 5, k = 2 * 5 = 10. Therefore, the correct choice is E. 10.

Edo invests $10,000 in an account earning 3% interest per annum compounding monthly. The value, V_n , of Edo’s investment after n months is given by

• V_n = 10000 x 1.0025ⁿ

• V_n = 10000 x 1.003ⁿ

• V_n = 10000 x 1.03ⁿ

• V_n = 10000 x 1.003¹²ⁿ

• V_n= 10000 x 1.03¹²ⁿ

Answer explanation

The correct choice is V_n = 10000 x 1.0025ⁿ because the interest is compounded monthly, so the growth factor is 1 + (0.03/12) = 1.0025.

Edo invests $10,000 in an account earning 3% interest per annum compounding monthly.

The effective interest rate for Edo’s investment is closest to

• 2.96%

• 2.98%

• 3.00%

• 3.02%

• 3.04%

Answer explanation

The effective interest rate for Edo’s investment is closest to 3.04% as it matches the given answer choice.

• 0

• 1

• 2

• 3

• 4

Todd invested $450,000 in an annuity at the start of 2024. The interest rate for this annuity is 3.75% per annum compounding monthly. He will receive regular monthly payments for the 15-year life of the annuity. In which year will the balance of the annuity first fall below $350,000?

• 2027

• 2028

• 2029

• 2030

• 2031

Answer explanation

The balance will fall below $350,000 in 2029 as the monthly payments gradually reduce the balance over time.

Jarryd invested $14,000 into an account earning compound interest at a fixed rate per time period. The graph below shows the balance of the account for four of the first five time periods after the initial investment. The information for time period 3 is not shown. Immediately after the interest was calculated for time period 3, Jarryd added an extra one-off amount

into the account.

This amount was closest to

• A. $224.03

• B. $225.97

• C. $228.62

• D. $229.38

• E. $231.46