Miskatonic University is a small liberal arts university in Arkham, Massachusetts. Each year since its inception in 1626, it has admitted exactly N freshman students, and exactly T transfer students at the sophomore, junior and senior levels. The fraction of freshman students surviving to their sophomore year is a, the fraction of sophomore students surviving to their junior year is b, and the fraction of junior students surviving to their senior year is c. A fraction d of seniors remains at the university for another year. Thus, the initial student population vector was v(0)-(N, T, T, T) and the population vector vi) satisfies v(t Av) + (0), where A is the class stratification matrix. a. Find the matrix A (4 pts) b. Use the equation ) Av()(0) recursively to show that (4 pts) Taking K-zd", show that 4-0 10 0 0) ΣÎ" 4 pts) ab b 1 0 Setting N-888, T-72, a- 5/8, b-3/4, c-7/8, and d-1/8, find the number of students at each level of their education, rounded off to the nearest whole student (the usual practice at Miskatonic University, where fractional students are common) d. (4 pts)
