Math Help - Average Rent?

Question

If I rent 3,000 square feet for $50,000 per year, and the rent increase 2% per year, what is an easy way to figure out the total rent and the average yearly rent?

Answer 1

Let n be the number of years over which the 2% increase applies. We will assume the increase compounds; that is, the 2% applies to the current year's rent, not to the original $50,000. Thus the rent will be $50,000 the first year, $51,000 the second, $52,020 the next, and so on.

The sum of the rents is

S = 50,000 + 51,000 + 52,020 + ... + 50,000(1.02)^(n-1)
= 50,000(1.02)^0 + 50,000(1.02)^1 + 50,000(1.02)^2 + ... + 50,000(1.02)^(n-1)

This is a geometric series. The sum of a geometric series can be found in math text books, not finance books, because as a finance question, the sum of the rents is irrelevant. In finance, you always have to take account of the time value of money when dealing with a stream of payments. The sum of the rents is

Sum = 50000(1 .02^n - 1) / (.02) <-- Helmut agrees with this

The average rent, also a meaningless number, is just the Sum divided by n.

Answer 2

Let R = rent
i = rate of increase
n = number of years
S = total rent
A = average rent
Then
R(n) = R(1 + i)^n, n starting at 0
S = R∑(1 + i)^k, from k = 0 to k = n - 1
S = (R)((1 + i)^n - 1)/(1 + i - 1)

S = R((1 + i)^n - 1)/ i
and
A = S/n = R((1 + i)^n - 1)/ ni

Plugging in the numbers,
S = $50,000(1.02^n - 1)/0.02
A = $50,000(1.02^n - 1)/(0.02n)

Answer 3

Ok the total rent for the next year should be
2/100*50000 + 50000 = $51000
then for every year after just do the same thing
2/100*51000 + 51000 = $52020 - year two