2.31. (Capital After Spending, Inflation, and Interest) Consider the following setup: - Begin with an initial capital C(0) in an interest-bearing account and let C(n) be the remaining capital at the end of the nth year. - Assume an interest rate r is applied at the end of each year to the capital remaining on that date. - At the end of the first year, assume that an amount S was spent from C(0) on goods and services, and money will be spent on similar goods and services in each of the subsequent years. - Suppose that the amount spent at the end of any specific year is the total amount spent by the end of the first year increased in subsequent years at the annual inflation rate i compounding annually until the end of the spec- ified year. Assume that r > i since investors are not interested in a market interest rate that is below the inflation rate. a) Show that the total capital at the end of the (n + 1)st year can be expressed recursively as follows in terms of the capital at the end of the previous year, taking into account spending, inflation, and interest growth: C(n+1)= (1+r) (C(n) - (1+i)"S). (2.80) b) Use induction to show that C(n) = (1+x)" - 1*19+(4+) (1+1)* s