Homework 2 Solutions

Homework 2 Solutions - Solutions to Homework Two CS 191 :...

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Solutions to Homework Two CS 191 : Discrete Structures I Jan. 30 , Winter 2007 Due Date: Feb. 13, Tuesday, before the class meeting. Section 1.7, p60 In Exercises 1-11, using induction, verify that each equation is true for every positive integer n. Exercise (6) 1 3 + 2 3 + 3 3 + … + n 3 = + 2 ) 1 ( 2 n n Answer : The inductive proof is as follows. Basis Step. When n = 1, the left side of the equation is equal to 1 3 = 1. The right side of the equation is equal to + 2 ) 1 1 ( 1 2 = 1. Therefore the equation is true for n =1. Inductive Step. Assume that the equation is true for n = k , where k 1. We shall prove that the equation will be true for n = k +1 also. Let n = k +1. Starting from the left side of the equation, we have the following derivations: 1 3 + 2 3 + 3 3 + … + k 3 + ( k +1) 3 = [1 3 + 2 3 + 3 3 + … + k 3 ] + ( k +1) 3 = + 2 ) 1 ( 2 k k + ( k +1) 3 //by induction! = 4 ) 1 ( 2 2 + k k + 4 ) 1 ( 4 3 + k = 4 )] 1 ( 4 [ ) 1 ( 2 2 + + + k k k = 4 ) 4 4 ( ) 1 ( 2 2 + + + k k k 1
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4 ) 2 ( ) 1 ( 2 2 + + k k = + + 2 ) 2 )( 1 ( 2 k k Therefore, the equation is also true for n = k +1. Inductive proof completes. In Exercise 21-24, use induction to prove the statement. Exercise (22) 11 n – 6 is divisible by 5, for all n 1. Answer : The inductive proof is as follows. Basis Step. When n = 1, 11 1 – 6 = 5 which is obviously divisible by 5. Therefore the statement is true for n =1. Inductive Step. Assume that the statement is true for n = k , where k 1. That is, 11 k – 6 is divisible by 5. We shall prove that the statement will be true for n = k +1 also. Let n = k +1. We have the following derivations: 11 n – 6 = 11 k +1 – 6 = 11 × 11 k – 6 = (10+1) 11 k – 6 = 10 × 11 k + 11 k – 6 = 10 × 11 k + (11 k – 6) = 5 × 2 × 11 k + (11 k – 6) Because 5 × 2 × 11 k is divisible by 5 and by induction, (11 k – 6) is divisible by 5, 11 k +1 – 6 is divisible by 5. Therefore, the statement is true for n = k +1 also. Inductive proof completes. 2
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Homework 2 Solutions - Solutions to Homework Two CS 191 :...

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