1) [24 Marks, 4 marks for each part] A rumour is introduced into the population of a small town. The proportion p, of the population that has been exposed to the rumur after n weeks after it was introduced is given by the recursively defined formula p1 = .05 and Patl = P. + .1p,(1 -P.). a) What proportion of the population will have been exposed to the rumour 5 weeks after it was introduced? (Choose 1) i) 0.0656 ii) 0.0717 iii) 0.783 My choice is: b) How many weeks will it take for at least 40% of the population to be exposed? (Choose 1) i) 28 ii) 35 iii) 215 My choice is: c) Use our Arithmetic Rules for Limits of Sequences to show that if {p, } converges with lim pr = n-+00 L, then either _ = 0 or L = 1. d) Prove using Mathematical Induction that for each n e N 0 Pn for each n E N. 2 f) Use what you know from parts c), d) and e) above to show that {p, } converges and find its limit