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?
5% growth is just
1200000*1.05^n after n years
for #c, we have
1200000*(1.05)^7*(0.92)^8 = 866,581
a. To determine the value of the building 7 years after it is purchased using a recursive formula, we need to use the information given:
Let V(n) represent the value of the building at the end of year n. The value at the end of year 1 is $1,200,000.
We can use the recursive formula V(n) = V(n-1) + 0.05 * V(n-1) to calculate the value at the end of each subsequent year.
Using this formula, we can calculate the value of the building 7 years after it is purchased:
V(1) = $1,200,000 (given)
V(2) = V(1) + 0.05 * V(1)
V(3) = V(2) + 0.05 * V(2)
V(4) = V(3) + 0.05 * V(3)
V(5) = V(4) + 0.05 * V(4)
V(6) = V(5) + 0.05 * V(5)
V(7) = V(6) + 0.05 * V(6)
Therefore, to find the value of the building 7 years after it is purchased using a recursive formula, you would need to calculate all the intermediate values using the given recursive relation.
b. To find the value of the building 4 years after it is purchased using an explicit formula, we can use the information provided:
Let V(n) represent the value of the building at the end of year n. The value at the end of year 1 is $1,200,000.
We can use the explicit formula V(n) = V(1) * (1 + 0.05)^n to calculate the value at the end of any particular year.
Using this formula, we can calculate the value of the building 4 years after it is purchased:
V(4) = $1,200,000 * (1 + 0.05)^4
Therefore, to find the value of the building 4 years after it is purchased using an explicit formula, you would need to substitute the values provided into the formula and simplify.
c. After the eighth year, the building starts depreciating at a rate of 8% per year. To find its value after the 15th year, we need to consider the decrease in value.
Let V(n) represent the value of the building at the end of year n.
From the eighth year onwards, the value decreases by 8% each year. This means that the value at the end of each year can be calculated using the formula V(n) = V(n-1) - 0.08 * V(n-1).
Using this formula, we can calculate the value of the building after the 15th year:
V(8) = V(7) - 0.08 * V(7)
V(9) = V(8) - 0.08 * V(8)
V(10) = V(9) - 0.08 * V(9)
V(11) = V(10) - 0.08 * V(10)
V(12) = V(11) - 0.08 * V(11)
V(13) = V(12) - 0.08 * V(12)
V(14) = V(13) - 0.08 * V(13)
V(15) = V(14) - 0.08 * V(14)
Therefore, to find the value of the building after the 15th year, you would need to calculate all the intermediate values using the given recursion for the depreciation formula.