Final answer:
The value of the machine at the end of the sixth year is $73,509.89.
Step-by-step explanation:
This problem involves the concept of depreciation, which is the decrease in value of an asset over time. Since the machine depreciates by 1/20th of its previous value every year, we can use a geometric sequence to find its value.
Let's break down the problem step by step:
1. The original cost of the machine is $100,000.
2. In the first year, the machine depreciates by 1/20th of its previous value:
- Value at the end of Year 1 = $100,000 - ($100,000 * 1/20) = $100,000 - $5,000 = $95,000
3. In the second year, the machine depreciates by 1/20th of its previous value:
- Value at the end of Year 2 = $95,000 - ($95,000 * 1/20) = $95,000 - $4,750 = $90,250
4. We can continue this pattern for each subsequent year until we reach the end of the sixth year:
- Value at the end of Year 3 = $90,250 - ($90,250 * 1/20) = $90,250 - $4,513 = $85,738
- Value at the end of Year 4 = $85,738 - ($85,738 * 1/20) = $85,738 - $4,287 = $81,451.40
- Value at the end of Year 5 = $81,451.40 - ($81,451.40 * 1/20) = $81,451.40 - $4,072.57 = $77,378.83
- Value at the end of Year 6 = $77,378.83 - ($77,378.83 * 1/20) = $77,378.83 - $3,868.94 = $73,509.89
Thus, the value of the machine at the end of the sixth year is $73,509.89.
The depreciation of the machine follows a geometric sequence because each year's value is a constant fraction (1/20th) of the previous year's value. The formula for the nth term of a geometric sequence is:
an = a1 * r(n-1)
Where an is the value at the end of the nth year, a1 is the initial value ($100,000), r is the common ratio (1 - 1/20 = 19/20), and n is the number of years.
In this case, we are interested in the value at the end of the sixth year (n = 6), so we can plug these values into the formula:
a6 = $100,000 * (19/20)(6-1) = $73,509.89