MATH 239 Spring 2010 Assignment 3 Due: May 28, 2010, 11:45AM EDT 1. In the Kingdom of Shallitland, King Jeffrey XVI declared that citizens of his coun-try (the Shallites) must use coins of these denominations: 1 byte, 7 bytes, 57 bytes, 80 bytes. Let n be a positive integer. (a) (2 marks) Assuming an endless supply of coins available in each denomina-tion, determine the number of ways that a collection of Shallite coins can make up n bytes. Write this as the coefﬁcient in a rational function. (b) (2 marks) King Jeffrey XVI ordered the 7-byte coins to be made of expensive silicon, and made a restriction that each citizen may possess at most 8 of these 7-byte coins. Given this restriction, determine the number of ways that one Shallite citizen can use their own coins to make up n bytes. Write this as the coefﬁcient in a rational function. (c) (2 marks) Let a n and b n be the answers for parts (a) and (b) respectively. Using your answers in (a) and (b), show that a n ≥ b n for all n
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Rational function, Recurrence relation, King Jeﬀrey XVI