Sequences and series

1. Assume that you invest an amount of money So at a nominal annual interest rate r. This means informally, that you get an extra amount of money rS after a year. However, this interest may be compounded several times a year: for instance, if the interest is compounded monthly, this means that after a month, you will get rSo/12. The next month however, you will get a proportion r/12 of what you have on the account at this time, namely So+rSo/12, and will therefore end up with an amount rSo r So + + iz (so +"). So rso 12 12 12 (a) Assume that you initially invest an amount S and that the interests are com- pounded n times per year. Explain quickly why, have t years, you will have an amount So r S = S. (1+)". n (b) Assume that you invest $1000 at rate of 5% for 5 years. Compute St when the interests are compounded annually / monthly / daily. Round to the nearest integer. (c) It is mathematically easier to consider interest compounded continuously, which means compounded every very small time interval. Give a precise interpretation of this concept, then compute St in this case. (d) Is it better to get a rate of 4.9% compounded continuously, or of 5% com- pounded annually