Real Estate an office building purchased for $1, 200,000 is appreciating because of rising property values in the city. At the end of each year its value is 105% of its value at the end of the previous year.
a. Use a recursive formula to determine what the value of the building will be 7 years after it is purchased.
b. Use an explicit formula to find the value of the building 4 years after it is purchased.
c. During the eighth year, the building begins to decrease in value at a rate of 8% per year. What would its value be after the 15th year?
a. Recursive formula:
Let V(n) represent the value of the building after n years.
V(n) = V(n-1) * 1.05
V(0) = $1,200,000
To find the value of the building 7 years after it is purchased, we need to calculate V(7) recursively:
V(1) = V(0) * 1.05 = $1,200,000 * 1.05 = $1,260,000
V(2) = V(1) * 1.05 = $1,260,000 * 1.05 = $1,323,000
V(3) = V(2) * 1.05 = $1,323,000 * 1.05 = $1,389,150
V(4) = V(3) * 1.05 = $1,389,150 * 1.05 = $1,458,607.50
V(5) = V(4) * 1.05 = $1,458,607.50 * 1.05 = $1,531,537.88
V(6) = V(5) * 1.05 = $1,531,537.88 * 1.05 = $1,608,114.78
V(7) = V(6) * 1.05 = $1,608,114.78 * 1.05 = $1,688,520.52
Therefore, the value of the building 7 years after it is purchased is $1,688,520.52.
b. Explicit formula:
The explicit formula for the value of the building after n years can be derived from the recursive formula.
V(n) = V(0) * (1.05)^n
Using this explicit formula, we can find the value of the building 4 years after it is purchased:
V(4) = $1,200,000 * (1.05)^4 = $1,200,000 * 1.2155 = $1,458,600
Therefore, the value of the building 4 years after it is purchased is $1,458,600.
c. After the eighth year, the value of the building is decreasing at a rate of 8% per year. This means that its value will be multiplied by 0.92 each year.
To find the value of the building after the 15th year, we need to calculate recursively:
V(8) = V(7) * 0.92 = $1,688,520.52 * 0.92 = $1,553,372.48
V(9) = V(8) * 0.92 = $1,553,372.48 * 0.92 = $1,428,936.28
V(10) = V(9) * 0.92 = $1,428,936.28 * 0.92 = $1,315,459.94
V(11) = V(10) * 0.92 = $1,315,459.94 * 0.92 = $1,211,800.54
V(12) = V(11) * 0.92 = $1,211,800.54 * 0.92 = $1,117,096.49
V(13) = V(12) * 0.92 = $1,117,096.49 * 0.92 = $1,030,241.31
V(14) = V(13) * 0.92 = $1,030,241.31 * 0.92 = $950,914.52
V(15) = V(14) * 0.92 = $950,914.52 * 0.92 = $877,856.33
Therefore, the value of the building after the 15th year is $877,856.33.
a. To determine the value of the building 7 years after it is purchased using a recursive formula, we need to start with the initial value of $1,200,000 and apply the appreciation rate of 105% each year.
Let's denote the value of the building at the end of year n as V_n.
The recursive formula can be written as: V_n = V_(n-1) * 1.05
Using this formula, we can calculate the value of the building year-by-year:
V_1 = $1,200,000 * 1.05 = $1,260,000
V_2 = $1,260,000 * 1.05 = $1,323,000
V_3 = $1,323,000 * 1.05 = $1,389,150
V_4 = $1,389,150 * 1.05 = $1,458,607.50
V_5 = $1,458,607.50 * 1.05 = $1,531,538.88
V_6 = $1,531,538.88 * 1.05 = $1,607,116.82
V_7 = $1,607,116.82 * 1.05 = $1,687,472.66
Therefore, the value of the building 7 years after it is purchased will be approximately $1,687,472.66.
b. To find the value of the building 4 years after it is purchased using an explicit formula, we can directly calculate the value using the formula:
V_n = V_0 * (1 + r)^n
Where V_0 is the initial value of the building ($1,200,000), r is the appreciation rate (1.05), and n is the number of years (4).
V_4 = $1,200,000 * (1 + 0.05)^4
= $1,200,000 * 1.05^4
= $1,200,000 * 1.21550625
≈ $1,458,607.50
Therefore, the value of the building 4 years after it is purchased will be approximately $1,458,607.50.
c. The value of the building during the eighth year decreases at a rate of 8% per year. To find its value after the 15th year, we need to consider both the appreciation for the first 7 years and the depreciation for the remaining years.
Using the recursive formula, we have determined that the value of the building after 7 years is approximately $1,687,472.66.
From the 8th year onwards, the value decreases at a rate of 8% per year. To calculate the value after each year, we use the formula:
V_n = V_(n-1) * (1 - r)
Where V_n is the value at the end of year n, V_(n-1) is the value at the end of the previous year, and r is the depreciation rate (0.08).
Using this formula, we can calculate the value after each year as follows:
V_8 = $1,687,472.66 * (1 - 0.08) = $1,551,288.78
V_9 = $1,551,288.78 * (1 - 0.08) = $1,428,523.30
V_10 = $1,428,523.30 * (1 - 0.08) = $1,316,841.15
V_11 = $1,316,841.15 * (1 - 0.08) = $1,215,011.55
V_12 = $1,215,011.55 * (1 - 0.08) = $1,121,811.08
V_13 = $1,121,811.08 * (1 - 0.08) = $1,036,208.73
V_14 = $1,036,208.73 * (1 - 0.08) = $957,261.53
V_15 = $957,261.53 * (1 - 0.08) = $884,144.89
Therefore, the value of the building after the 15th year will be approximately $884,144.89.