Here is the question (Course 2 Exam - May/2000/ Q #37).... A customer is offered an investment where interest is calculated according to the following force of interest: .02t Between time 0 and 3 (inclusive) and .045 at time greater than 3.

The customer invests 1000 at time t=0. What nominal rate of interest, compounded quarterly, is earned over the first four-year period?

I believe i understand how to do it. I am just having a minor complication. I hope this makes sense.....

I want to discuss finding the a(4), using the force of interest given in the question. So what should be done is a(3) multiplied by the accumulated value of just the 4th year. Here is where I get confused.

The accumulated value of just the 4th year should be the a(4)/a(3). (Even if you use integrals). a(4) uses a different force of interest than a(3). So if i use integrals how does it make sense to say e^integral(3 to 4) of .045 dt? We are finding a(4) and then subtracting the three previous years. But the 3 previous years is measured with a different force of interest?!!

I hope this makes sense. Im sure there is a simple answer to this, but i am just confused. Thanks in advance for all your help.