99. Sierpinski triangle The fractal called the Sierpinski triangle is the limit of a sequence of figures. Starting with the equilateral tri- angle with sides of length 1, an inverted equilateral triangle with sides of length 1/2 is removed. Then, three inverted equilateral triangles with sides of length 1/4 are removed from this figure (see figure). The process continues in this way. Let T, be the total area of the removed triangles after stage n of the process. The area of an equilateral triangle with side length Lis A = V3L 4 a. Find 7, and 72, the total area of the removed triangles after stages 1 and 2, respectively. b. Find T, for n = 1, 2, 3, . ... c. Find lim T, d. What is the area of the original triangle that remains as n - co? Initial stage First stage Second stage 100. Multiplier effect Imagine that the government of a small com- munity decides to give a total of $ W, distributed equally, to all its citizens. Suppose each month each citizen saves a fraction p of his or her new wealth and spends the remaining 1 - p in the commu- nity. Assume no money leaves or enters the community, and all the spent money is redistributed throughout the community. a. If this cycle of saving and spending continues for many months, how much money is ultimately spent? Specifically, by what factor is the initial investment of $W increased? Economists refer to this increase in the investment as the multiplier effect. b. Evaluate the limits p - 0 and p - 1, and interpret their mean- ings. (See the Guided Project Economic stimulus packages for more on stimulus packages.)