A rumour is being spread by students in Melbourne. At the start of day 1, a set of 10 students know the rumour. Let So be this set of students, and for n 1, let Sn denote the set of students who are told the rumour during day n. Also, write Sn to stand for the number of students in Sn. Assume that students in Sn pass on the rumour on day n+1 to the following numbers of new students: 0.2s, in cafes, 0.3s in public transport, 0.4s, at work, and 0.38, in university classes. Assume that no student is told the rumour twice, and students in S do not pass on the rumour any more after day n+1. In the following, assume the above specifications hold precisely, ignoring the fact that the numbers of students of each type should be integers. (a) For every integer n 1, derive a formula for s using exponential notation. (b) Show how to use logarithms to find the first value of n for which sn exceeds 10,000. (c) The government is embarassed by the rumour, so it introduces social distancing laws at the end of day k, to reduce the spread of the rumour. This restriction is only imposed in public transport and in university classes. As a result, the number of students told the rumour from these two sources changes to 0.1sn each (on day n + 1, when n k, as described above). The other sources follow the same rule as before. Find the largest value of k that will permit 850 to be less than 10.