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Recursion, GEN1 2024 NHT 17 MC

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

\(M_0=60 \, 000, \quad M_{n+1}=0.85 \ M_n\)

An equivalent rule to determine the value of the car after \(n\) years is

  1. \(M_n=60\,000-0.85 n\)
  2. \(M_n=60\,000+0.85 n\)
  3. \(M_n=60\,000+0.85^n\)
  4. \(M_n=60\,000 \times 0.85^{n-1}\)
  5. \(M_n=60\,000 \times 0.85^n\)
Show Answers Only

\(E\)

Show Worked Solution

\(\text{Reducing balance}\ \ \Rightarrow \ \ \text{Eliminate A, B and C}\)

\(\text{After 1 year, value}\ M_1=60\,000 \times 0.85\)

\(\Rightarrow E\)

CORE, GEN1 2019 NHT 18 MC

A truck was purchased for $134 000.

Using the reducing balance method, the value of the truck is depreciated by 8.5% each year.

Which one of the following recurrence relations could be used to determine the value of the truck after `n` years, `V_n`?

  1. `V_0 = 134\ 000,quadqquad V_(n + 1) = 0.915 xx V_n`
  2. `V_0 = 134\ 000,quadqquad V_(n + 1) = 1.085 xx V_n`
  3. `V_0 = 134\ 000,quadqquad V_(n + 1) = V_n - 11\ 390`
  4. `V_0 = 134\ 000,quadqquad V_(n + 1) = 0.915 xx V_n - 8576`
  5. `V_0 = 134\ 000,quadqquad V_(n + 1) = 1.085 xx V_n - 8576`
Show Answers Only

`A`

Show Worked Solution

`text(Each year, value decreases by 8.5%)`

`:. V_1` `= V_0 – 0.085 xx V_0`
  `= 0.915 xx V_0`

 
`=>\ A`

CORE, FUR1 2019 VCAA 19 MC

Geoff purchased a computer for $4500. He will depreciate the value of his computer by a flat rate of 10% of the purchase price per annum.

A recurrence relation that Geoff can use to determine the value of the computer after `n` years, `V_n`, is

  1. `V_0 = 4500, qquad quad V_(n + 1) = V_n - 450`
  2. `V_0 = 4500, qquad quad V_(n + 1) = V_n + 450`
  3. `V_0 = 4500, qquad quad V_(n + 1) = 0.9 V_n`
  4. `V_0 = 4500, qquad quad V_(n + 1) = 1.1 V_n`
  5. `V_0 = 4500, qquad quad V_(n + 1) = 0.1 (V_n - 450)`
Show Answers Only

`A`

Show Worked Solution

`text(Flat rate depreciation each year)`

`= 10% xx 4500`

`= $450`

`:. V_(n + 1) = V_n – 450`

`=>  A`

CORE, FUR2 2018 VCAA 5

After three years, Julie withdraws $14 000 from her account to purchase a car for her business.

For tax purposes, she plans to depreciate the value of her car using the reducing balance method.

The value of Julie’s car, in dollars, after `n` years, `C_n`, can be modelled by the recurrence relation shown below

`C_0 = 14\ 000, qquad C_(n + 1) = R xx C_n`

  1. For each of the first three years of reducing balance depreciation, the value of `R` is 0.85

     

    What is the annual rate of depreciation in the value of the car during these three years?  (1 mark)

  2. For the next five years of reducing balance depreciation, the annual rate of depreciation in the value of the car is changed to 8.6%.

     

    What is the value of the car eight years after it was purchased?
    Round your answer to the nearest cent.  (2 marks)

Show Answers Only
  1. `15 text(%)`
  2. `$5484.23\ text{(nearest cent)}`
Show Worked Solution

a.  `text(S)text(ince)\ \ R = 0.85,`

COMMENT: Note almost half of students answered incorrectly here!

`=> 85 text(% of the car’s value remains at the end of each)`

      `text{year (vs the value at the start of the same year.)}`

`:.\ text(Annual rate of depreciation) = 15 text(%)`

  
b.   `text(Value after 3 years)`

♦ Mean mark 42%.

`C_3` `= (0.85)^3 xx 14\ 000`
  `= $8597.75`

 
`:.\ text(Value after 8 years)`

`C_8` `= (0.914)^5 xx C_3`
  `= (0.914)^5 xx 8597.75`
  `= $5484.23\ text{(nearest cent)}`

CORE, FUR2 2017 VCAA 5

Alex is a mobile mechanic.

He uses a van to travel to his customers to repair their cars.

The value of Alex’s van is depreciated using the flat rate method of depreciation.

The value of the van, in dollars, after `n` years, `V_n`, can be modelled by the recurrence relation shown below.

`V_0 = 75\ 000 qquad V_(n + 1) = V_n - 3375`

  1. Recursion can be used to calculate the value of the van after two years.

     

    Complete the calculations below by writing the appropriate numbers in the boxes provided.   (2 marks)


    1. By how many dollars is the value of the van depreciated each year?   (1 mark)
    2. Calculate the annual flat rate of depreciation in the value of the van.
    3. Write your answer as a percentage.   (1 mark)
  3. The value of Alex’s van could also be depreciated using the reducing balance method of depreciation.
  4. The value of the van, in dollars, after `n` years, `R_n`, can be modelled by the recurrence relation shown below.

     

            `R_0 = 75\ 000 qquad R_(n + 1) = 0.943R_n`

    At what annual percentage rate is the value of the van depreciated each year?   (1 mark)

Show Answers Only

a.

b.i.  `$3375`

b.ii. `4.5text(%)`

c.  `5.7text(%)`

Show Worked Solution

a.   

  
b.i.   `$3375`
  

b.ii.    `text(Annual Rate)` `= 3375/(75\ 000) xx 100`
    `= 4.5text(%)`

 

c.    `text(Annual Rate)` `= (1-0.943) xx 100text(%)`
    `= 5.7text(%)`

CORE, FUR2 SM-Bank 4

Damon runs a swim school.

The value of his pool pump is depreciated over time using flat rate depreciation.

Damon purchased the pool pump for $28 000 and its value in dollars after `n` years, `P_n`, is modelled by the recursion equation below:

`P_0 = 28\ 000,qquad P_(n + 1) = P_n - 3500`

  1. Write down calculations, using the recurrence relation, to find the pool pump's value after 3 years.   (1 mark)

  2. After how many years will the pump's depreciated value reduce to $7000?   (1 mark)

The reducing balance depreciation method can also be used by Damon.

Using this method, the value of the pump is depreciated by 15% each year.

A recursion relation that models its value in dollars after `n` years, `P_n`, is:

`P_0 = 28\ 000, qquad P_(n + 1) = 0.85P_n`

  1. After how many years does the reducing balance method first give the pump a higher valuation than the flat rate method in part (a)?   (2 marks)

Show Answers Only
  1. `$17\ 500`
  2. `6\ text(years)`
  3. `4\ text(years)`
Show Worked Solution
a.    `P_1` `= 28\ 000-3500 = 24\ 500`
  `P_2` `= 24\ 500-3500 = 21\ 000`
  `P_3` `= 21\ 000-3500 = 17\ 500`

  
`:.\ text(After 3 years, the pump’s value is $17 500.)`
  

b.   `text(Find)\ n\ text(such that:)`

`7000` `= 28\ 000-3500n`
`3500n` `= 21\ 000`
`n` `= (21\ 000)/3500`
  `= 6\ text(years)`

  
c.   `text(Using the reducing balance method)`

`P_1` `= 0.85 xx 28\ 000 = 23\ 800`
`P_2` `= 0.85 xx 23\ 800 = 20\ 230`
`P_3` `= 0.85 xx 20\ 230 = 17\ 195`
`P_4` `= 0.85 xx 17\ 195 = 14\ 615.75`

  
`text{Using the flat rate method (see part (a))}`

`P_4 = 17\ 500-3500 = 14\ 000`

`14\ 615.75 > 14\ 000`
  

`:.\ text(After 4 years, the reducing balance method)`

`text(first values the pump higher.)`

CORE, FUR2 SM-Bank 3

Luke purchased a new pizza oven for his restaurant for $23 500.

He can depreciate the pizza oven using the reducing balance method at a rate of 12.5% per year.

  1. If `V_n` represents the value of the pizza oven after `n` years, write a recurrence relation that models its value.   (1 mark)

  2. During what year will the pizza oven's value drop below $15 000?   (1 mark)

Luke has been advised that he can use flat rate depreciation at 10% of the purchase price.

  1. After 4 years, show which depreciation method gives the pizza oven the highest value?   (1 mark)

Show Answers Only
  1. `V_0 = 23\ 500,qquadV_(n + 1) = 0.875V_n`
  2. `text(year 4)`
  3. `text(See Worked Solutions)`
Show Worked Solution
a.    `V_0` `= 23\ 500`
  `V_1` `= 23\ 500-(12.5text(%) xx 23\ 500)`
    `= 0.875 V_0`
  `V_2` `= 0.875(0.875V_0)`
    `= 0.875 V_1`

  
`:.\ text(Recurrence relationship:)`

`V_0 = 23\ 500,qquadV_(n + 1) = 0.875V_n`
  

b.    `V_1` `= 0.875 xx 23\ 500 = 20\ 562.50`
  `V_2` `= 0.875 xx 20\ 562.50 = 17\ 992.1875`
  `V_3` `= 0.875 xx 17\ 992.1875 = 15\ 743.16…`
  `V_4` `= 0.875 xx 15\ 743.16… = 13\ 775.26…`

  
`:.\ text(The value drops below $15 000 in year 4.)`

  
c.   `text(Value after 4 years using reducing balance)`

`= 13\ 775.26`

`text(Depreciation each year for flat rate)`

`= 10text(%) xx 23\ 500`

`= $2350`

`text(Value of pizza oven after 4 years,)`

`= 23\ 500-(4 xx 2350)`

`= $14\ 100`
 

`:.\ text(The flat rate depreciation method)`

`text(values the pizza oven highest.)`