In each part below, provide an expression for the value at 1/1/2010 of the payments described. Your expression may incorporate annuity symbols and/or other compound interest expressions. You need to provide an expression showing how the required interest rate is calculated without evaluating it. DO NOT evaluate your expression. (a) A perpetuity making continuous payments at the rate of $4,000 p.a. Payments commence on 1/1/2016. The interest rate is 4% per half-year effective. (3 marks) (b) An annuity making quarterly payments of $25 from 1/1/2011 to 1/1/2020 inclusive. The interest rate is 10% p.a. effective. (3 marks) (C) An annuity making annual payments from 1/1/2015 to 1/1/2025 inclusive. The payments reduce in arithmetic progression, the first payment being $1,000 and the last payment being $10. The force of interest 8% p.a. (4 marks) (d) A continuous annuity making payments from 1/1/2005 to 1/1/2010. Within this range, the rate of payment t years after 1/1/2005 is $(100+20t) p.a. The interest rate is 10% p.a. compounded half-yearly. (4 marks)