A house is expected to have a maintenance cost of 1000 the f

A house is expected to have a maintenance cost of $1000 the first year. It is believed that the maintenance cost will increase $500 per year. The interest rate is 6% compounded annually. Over a 10-year period, what will be the approximate effective annual maintenance cost? 1.$1900 2.$3000 3.$3500 4.$3800

Solution

The cost of maintenanceof the house in the 1st year is $ 1000. The cost of maintenanceof the house in the 2nd, 3rd, 4th years are $ (1000 + 500), $ ( 1000 + 500 + 500), $( 1000 + 500 + 500) etc. This is an arithmetic series. We know that the nth term of an arithmetic series whose 1st term is a and the common difference is b is,a +( n-1)b . Here a = 1000, b = 500 and n = 10 so that the maintenance cost in the 10th year is 1000 + (10 - 1) 500 = $ 1000 + 4500 = $ 5500.

However, if the question is about the total cost of the maintenance of the house over a 10 year perio, then we know that the sum of a finite arithmetic series is S = n/2 { 2a + (n -1)b] . Here n = 10, a = 1000 and b = 500 so that S = 10/2 [ 2*1000 + ( 10 - 1) 500] = 5 ( 2000+ 9*500) = 5 ( 2000+ 4500) = 5( 6500) = $ 32500.

NOTE:

The relevance of quoting a rate of interest is not clear.