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?
f(0) = 1200000
f(t+1) = 1.05 f(t)
so,
f(t) = f(0) * 1.05^t
and so on
To determine the value of the office building over a period of time, we can use the given information to create a recursive formula and an explicit formula.
a. Recursive formula:
The recursive formula represents the value of the building at year n in terms of its value in the previous year.
Let V(n) represent the value of the building at year n.
We know that at the end of each year, the value increases by 105% of its previous value.
Therefore, the recursive formula can be written as:
V(n) = 1.05 * V(n-1)
To find the value 7 years after the building is purchased, we can use this recursive formula.
V(1) = $1,200,000 (initial value)
Substituting into the formula:
V(2) = 1.05 * V(1) = 1.05 * $1,200,000 = $1,260,000
V(3) = 1.05 * V(2) = 1.05 * $1,260,000 = $1,323,000
V(4) = 1.05 * V(3) = 1.05 * $1,323,000 = $1,389,150
V(5) = 1.05 * V(4) = 1.05 * $1,389,150 = $1,458,608.75
V(6) = 1.05 * V(5) = 1.05 * $1,458,608.75 = $1,531,539.19
V(7) = 1.05 * V(6) = 1.05 * $1,531,539.19 = $1,608,116.15
Therefore, the value of the building 7 years after purchase would be approximately $1,608,116.15.
b. Explicit formula:
The explicit formula represents the value of the building at year n directly, without relying on previous values.
The value of the building at year n can be calculated using the initial value and the rate of appreciation.
The explicit formula is given by:
V(n) = V(1) * (1.05)^n
To find the value 4 years after the building is purchased, we can use this explicit formula.
V(4) = $1,200,000 * (1.05)^4
Solving:
V(4) = $1,200,000 * (1.05)^4 = $1,464,300
Therefore, the value of the building 4 years after purchase would be $1,464,300.
c. During the eighth year, the building decreases in value by 8% per year.
We need to calculate the value of the building after the 15th year.
To determine the value after the 15th year, we need to make adjustments for both the appreciation in the first 7 years and the depreciation in the last 8 years.
From part a, we know the value after 7 years is $1,608,116.15.
To calculate the value after the 8th year, we need to account for the 8% depreciation.
V(8) = V(7) * (1 - 0.08) = $1,608,116.15 * (1 - 0.08) = $1,479,888.87
We then apply the 8% depreciation for the next 7 years:
V(9) = V(8) * (1 - 0.08) = $1,479,888.87 * (1 - 0.08) = $1,362,690.80
V(10) = V(9) * (1 - 0.08) = $1,362,690.80 * (1 - 0.08) = $1,253,485.66
V(11) = V(10) * (1 - 0.08) = $1,253,485.66 * (1 - 0.08) = $1,151,399.13
V(12) = V(11) * (1 - 0.08) = $1,151,399.13 * (1 - 0.08) = $1,055,592.13
V(13) = V(12) * (1 - 0.08) = $1,055,592.13 * (1 - 0.08) = $965,261.30
V(14) = V(13) * (1 - 0.08) = $965,261.30 * (1 - 0.08) = $879,633.24
V(15) = V(14) * (1 - 0.08) = $879,633.24 * (1 - 0.08) = $798,969.99
Therefore, the value of the building after the 15th year would be approximately $798,969.99.