# s4 - Selected Solutions to Homework 4 1 4 in Appendix C Let...

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Selected Solutions to Homework # 4 1. # 4 in Appendix C: Let r be a real number, r 6 = 1. Prove that for every n Z + , 1 + r + r 2 + ··· + r n - 1 = r n - 1 r - 1 . Proof: We give a proof using induction. Let r R and r 6 = 1. Let P n be the statement that the above equation holds for the integer n . The statement P 1 reads 1 = r - 1 r - 1 , which is evidently true. Suppose that the statement P n is true for some n 1. We are to show that the statement P n +1 is true. Consider 1 + 2 + ··· + r n - 1 + r n . By grouping the ﬁrst n terms and applying P n , we have 1+2+ ··· + r n - 1 + r n = r n - 1 r - 1 + r n = r n - 1 r - 1 + r n ( r - 1) r - 1 = r n +1 - 1 r - 1 . Thus we have shown that P n +1 is a consequence of P n . By the principle of mathematical induction, P n is true for every integer n 1. Q.E.D. 2. # 16 in Appendix C. This is the game known as the “Towers of Hanoi”. I will give a brief sketch of how to solve this problem.

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s4 - Selected Solutions to Homework 4 1 4 in Appendix C Let...

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