Question

A rule of thumb states that cars used for personal use depreciates by 15% each year (that is the value of the car is 15% less at the end of the year as it was at the beginning of the year). Suppose that a new car is purchased for $30,000. Let Yn be the value of the car after n years. Do the following:

(a) Create the difference equation that will model the cars value.

(b) Solve the difference equation.

(c) Use the solution of the difference equation to determine the value of the car after 5 years.

Answer 1

Part (a)

Value at the end of year, n = V_n = Value at the end of year (n - 1) - Depreciation rate x Value at the end of year (n - 1) = V_n-1 - V_n-1 x 15%

Hence, V_n = V_n-1 x (1 - 0.15)

Or, V_n = 0.85V_{n - 1}

Part (b)

Applyig this formula recurssively, we get:

V_n = 0.85V_{n - 1}

V_{n - 1} = 0.85V_{n - 2}

V_{n - 2} = 0.85V_{n - 3}

and so on....

Hence, we can substitute from the bottom most equation upward to get:

V_n = 0.85ⁿ x V₀ = 30,000 x 0.85ⁿ

Part (c)

n = 5

Hence, Value of the car after 5 years = V₅ = 30,000 x 0.85⁵ = $ 13,311.16