Question

80. You buy a new car for $24,000. At the end of n years, the value of your car is given by the sequence: an=24000(3/4)n, n=1, 2, 3, ... Find a5 and write a sentence explaining what this value represents. Describe the nth term of the sequence in terms of the value of the car at the end of each year.

Answer 1

The value at the end of the fifth year (a5) is equal to $5,696.

As the number of years n is the exponent of a number that is smaller than 1, the value of the car is decreasing with the years.

We also can conclude that the value of the car decreases 25% each year.

Explanation:

The value of the car is represented by an=24000(3/4)^n, where n is the year.

The value at the end of the fifth year (a5) is equal to:

`a_5=24,000\left((3)/(4)\right)^5=24,000*0.2373=5,695`

As the number of years n is the exponent of a number that is smaller than 1, the value of the car is decreasing with the years.

We also can conclude that the value of the car decreases 25% each year:

`a_n/a_(n-1)=(3/4)^(n-(n-1))=(3/4)^1=0.75`