Ok, so I have several questions on this topic area but am able to enter a number of answers for one question in particular yet achieve the same outcome (a score of 75%). This would suggest that the question I can enter several variations of answer for is incorrect, right?

Please tell me what or where I'm going wrong with this if you know as it's driving me crazy!

Question

The value of a machine which costs £35000 has a costs of repairs and maintenance of £2000 in the first year and increases by 10% per year.
After how many years will it be cheaper to buy a new machine than to pay for the repairs and maintenance up till then?

I have tried using the following answers:

31 years (2000 * 1.1^31 = £38388.68499)
30 (30.04 rounded to nearest whole number as 2000*1.1^30.04 = £35032.10693)
11 (using sum of geometric sequence S11 > £35000

As I said earlier all of these give me a 75% scoring for the questions. HELP! :(

Unless I'm missing something...

T_n = ar^{n-1}

35000 = 2000(1.1)^{n-1}

1.1^{n-1) = 17.5

n - 1 = \log_{1.1}(17.5)

n = 31.03

But since we want the year when it exceeds 35000, we take the nearest higher integer, which is 32 years.

Replacing in the equation, we get:

T_{31} = 2000(1.1)^{31-1} = 34898

T_{32} = 2000(1.1)^{32-1} = 38388

Hence, on the 31st year, it will still be cheaper to pay the repair and maintenance, but on the 32nd year, it's better to buy a new one.