the first month each property is acquired, it will house 2 tenants.
the number of tenants in each property will increase by 2 per month with a maximum of 20 tenants per tenants per property.
how many tenants will you have after 36 months.
after 36 months, there will be 36 properties.
each property will have 20 tenants after 10 months.
that would be the first month the property is purchased plus 9 months after.
2 + 9 * 2 = 2 + 18 = 20.
number 36 will have 2
number 35 will have 4
number 34 will have 6
number 33 will have 8
number 32 will have 10
number 31 will have 12
number 30 will have 14
number 29 will have 16
number 28 will have 18
number 27 will have 20
property numbers 1 through 27 will have 20 each.
this is the maximum that each property can have.
there are 9 properties left and those will form an arithmetic progression
the formula for an arithmetic progeression is An = A1 + (n-1) * d.
the formula for the sum of an arithmetic progression is Sn = n * (A1 + An) / 2
A1 is equal to 2.
d is equal to 2.
A9 = 2 + 8*2 = 2 + 16 = 18.
that equates to property number 28.
note that you are counting down from 36, but your sequence is counting up from 1.
that means that:
A1 = property number 36
A2 = property number 35
A3 = property number 34
A4 = property number 33
A5 = property number 32
A6 = property number 31
A7 = property number 30
A8 = property number 29
A9 = property number 28
property number 28 is the 9th property counting down from 36 and including property number 36.
36 - 29 = 8 + 1 to include 36 = 9
Sn is equal to 9 * (2 + 18) / 2 = 9 * 20/2 = 9 * 10 = 90
that equates to the number of tenants in property numbers 28 through 36.
the total number of tenants is therefore 27 * 20 + 90 = 540 + 90 = 630.
this can be seen in the following spreadsheet.
columns are months.
rows are months.
data in each cell is the number of tenants in the indicated month.
months 13 through 25 have been hidden from view so you could see what's happening in the first 12 months and in the last 11 months.
the bottom row shows the total number of tenants in each month.
