In a pyramid scheme, the members of a company recruit more members into the company. A person that has already been recruited cannot be recruited again. We will also assume that no one leaves the company once they are recruited.A person's level in the company refers to how many recruitment steps they are away from the founder.The founder is at level 0. The people recruited by the founder are at level 1. The recruits of the people at level 1 are at level 2, and so on.The height of the pyramid scheme is the maximum level of anyone in the company. A company may enforce a recruitment limit which is an upper limit on the number of people any individual in thecompany can recruit. The recruitment limit is a non-negative integer.Prove by induction that a company with a recruitment limit of m and height h has at most m^h people in the company who have yet to recruit anyone into the company.
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the hardship is to explain it in math language – Linda dadad Oct 12 '17 at 09:15
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I don't know what that means. Either way, if you've formulated in an informal method of proving this, you should put that in your question. Everything that you've tried should always be in the question, so that we can answer your question better. – Oct 12 '17 at 09:19
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"The recruitment limit is a non-negative integer." It seems here the recruitment limit should be a positive integer, as if the recruitment limit is 0 and the height is 0, then we have $0^0$. – Element118 Oct 12 '17 at 11:44
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If this is wrong feel free to down vote! It required a picture in the middle since I wasn't sure how to say it in maths only.
Z is the number of people that haven't recruited anyone.
Erin
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I believe this makes sense...but it's not really...math, you know.. – Linda dadad Oct 12 '17 at 09:15
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I think you're going to have to use words at some point - the question is all words, and you'll need to explain how that turns into maths. You just have to work out the way your teacher would like best. My answer should give you 90% of what you need. – Erin Oct 12 '17 at 09:18
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This answer seems to be assuming that all the members that have not recruited are at the same level. – Element118 Oct 12 '17 at 11:40
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@Element118 The question asks for the number "at most" - if there are some not on the same level you will have less than the max for that height. Each person on the second last row that hasn't recruited anyone only adds 1 to Z instead of m – Erin Oct 12 '17 at 12:13
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To make this rigorous, how about consider the set of people who did not recruit anyone yet in the height h+1 case, and then try removing them? – Element118 Oct 12 '17 at 12:17