1) [24 Marks, 4 marks for each part] A rumour is introduced into the population of a small town. The proportion p7; of the population that has been exposed to the rumur after in weeks after it was introduced is given by the recursively dened formula p1 : .05 and pnIl : p1; + -1pn(1 _ pm)- 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 {pH} converges with lim p" = Hi'DO L, then either L = 0 or L = 1. d) Prove using Mathematical Induction that for each n E N D p" for each n E N. f) Use what you know from parts c), d) and e) above to show that {pm} converges and nd its limit