Soln 4 - MACM 101 Discrete Mathematics I Exercises on...

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MACM 101 — Discrete Mathematics I Exercises on Induction and Combinatorics. Due: Monday, November 15th (at the beginning of the class) Reminder: the work you submit must be your own. Any collaboration and consulting outside resources must be explicitly mentioned on your submis- sion. 1. Prove that for every positive integer n 1 · 2 1 + 2 · 2 2 + 3 · 2 3 + ... + n · 2 n = ( n - 1)2 n +1 + 2 . Solution: We use induction. Let P ( n ) denote this equality for the integer n . Basis case. P (1) means the equality 1 · 2 = 1(1+1)(1+2) 3 , which is obvi- ously true. Inductive step. Suppose that P ( k ) is true, that is, 1 · 2 + 2 · 3 + 3 · 4 + ... + k · ( k + 1) = k ( k + 1)( k + 2) 3 . We have to prove P ( k + 1): 1 · 2+2 · 3+3 · 4+ ... + k · ( k +1)+( k +1) · ( k +2) = ( k + 1)( k + 2)( k + 3) 3 . We have 1 · 2 + 2 · 3 + 3 · 4 + ... + k · ( k + 1) + ( k + 1) · ( k + 2) = k ( k + 1)( k + 2) 3 + ( k + 1) · ( k + 2) = k ( k + 1)( k + 2) + 3( k + 1)( k + 2) 3 = ( k + 1)( k + 2)( k + 3) 3 1
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2. (The gossip problem.) Suppose there are n people in a group, each aware of a scandal no one else in the group knows about. These people communicate by telephone; when two people in the group talk, they share information about all scandals each knows about. The gossip problem asks for G ( n ) , the minimum number of telephone calls that are needed for all n people to learn about all the scandals. Prove that G ( n ) 2 n - 4 . Solution: We prove this statement for all n such that n 4. Denote by
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Soln 4 - MACM 101 Discrete Mathematics I Exercises on...

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