Chapter_04 - Chapter 4 1 Suppose you wish to prove that the...

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Chapter 4 1. Suppose you wish to prove that the following is true for all positive integers n by using the Principle of Mathematical Induction: 1 + 3 + 5 + ... + (2 n 1) = n 2 . (a) Write P (1) (b) Write P (72) (c) Write P (73) (d) Use P (72) to prove P (73) (e) Write P ( k ) (f) Write P ( k + 1) (g) Use the Principle of Mathematical Induction to prove that P ( n ) is true for all positive integers n Ans: (a) 1 = 1 2 . (b) 1 + 3 + 5 + ... + 143 = 72 2 . (c) 1 + 3 + 5 + ... + 145 = 73 2 . (d) 1 + 3 + 5 + ... + 145 = (1 + 3 + 5 + ... + 143) + 145 = 72 2 + 145 = 72 2 + 2 72 + 1 = (72 + 1) 2 = 73 2 . (e) 1 + 3 + ... + (2 k 1) = k 2 . (f) 1 + 3 + ... + (2 k + 1) = ( k + 1) 2 . (g) P (1) is true since 1 = 1 2 . P ( k ) P ( k + 1): 1 + 3 + ... + (2 k + 1) = k 2 + (2 k + 1) = ( k + 1) 2 . 2. Suppose you wish to use the Principle of Mathematical Induction to prove that 1 1 ! + 2 2 ! + 3 3 ! + ... + n n ! = ( n + 1) ! 1 for all n 1 . (a) Write P (1) (b) Write P (5) (c) Write P ( k ) (d) Write P ( k + 1) (e) Use the Principle of Mathematical Induction to prove that P ( n ) is true for all n 1 Ans: (a) 1 1 ! = 2 ! 1. (b) 1 1 ! + 2 2 ! + ... + 5 5 ! = 6 ! 1. (c) 1 1 ! + 2 2 ! + ... + k k ! = ( k + 1) ! 1. (d) 1 1 ! + 2 2 ! + ... + ( k + 1)( k + 1) ! = ( k + 2) ! 1. (e) P (1) is true since 1 1 ! = 1 and 2 ! 1 = 1. P ( k ) P ( k + 1): . 11 2 2 ( 1 )( 1 ) (1 )1 ) ) ) [ 1 ) ] 1 ) (2 ) 1 kk k k ⋅ !+ ⋅ !+. ..+ + + ! =+ ! ++ + ! ! ++− !+− =+! Page 55
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3. Use the Principle of Mathematical Induction to prove that 1 23 2(1 ) 1 122 2 (1 )2 3 nn + + + +...+ − = for all positive integers n . Ans: P (1): 2 3 −+ −= 12 , which is true since both sides are equal to 1. P ( k ) P ( k + 1): 11 21 1 1 1 2 ( 1) 1 2 ( 1 3( 2 122 ) 2 33 kk k k 1 1 + ++ +− + = + − = + 2 ( 1) (1 3( 1)) 1 2 ( 1) ( 2) 1 2 ( 1 k k + + −− + = 2 1 3 + == . 4. Use the Principle of Mathematical Induction to prove that 1 + 2 n 3 n for all n 1. Ans: P (1): 1 + 2 1 3 1 , which is true since both sides are equal to 3. P ( k ) P ( k + 1): 1 + 2 k + 1 = (1 + 2 k ) + 2 k 3 k + 2 k 3 k + 3 k = 2 3 k < 3 3 k = 3 k + 1 . 5. Use the Principle of Mathematical Induction to prove that n 3 > n 2 + 3 for all n 2. Ans: P (2): 2 3 > 2 2 + 3 is true since 8 > 7. P ( k ) P ( k + 1): ( k + 1) 2 + 3 = k 2 + 2 k + 1 + 3 = ( k 2 + 3) + 2 k + 1 < k 3 + 2 k + 1 k 3 + 3 k k 3 + 3 k 2 + 3 k + 1 = ( k + 1) 3 . 6. Use the Principle of Mathematical Induction to prove that 2 | ( n 2 + n ) for all n 0. Ans: P (0): 2 | 0 2 + 0, which is true since 2 | 0. P ( k ) P ( k + 1): ( k + 1) 2 + ( k + 1) = ( k 2 + k ) + 2( k + 1), which is divisible by 2 since 2 | k 2 + k and 2 | 2( k + 1). 7. Use the Principle of Mathematical Induction to prove that 1 31 92 7 3 2 n n + 13 + + + +...+ = for all n 0. Ans: P (0): 1 2 = 1 , which is true since 1 = 1. P ( k ) P ( k + 1): 1 2 3 3 3 3 22 k + + + + . . . += = 2 1 2 k + . Page 56
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8. Use the Principle of Mathematical Induction to prove that (3 1) 1471 0 ( 3 2 ) 2 nn n +++ + . . . + − = for all n 1. Ans: P (1): 12 2 = 1 , which is true since 1 = 1. P ( k ) P ( k + 1): 2 (3 14 ( 3 ( 1 )2 ) ( 3 1 ) 2 (3 2(3 3 5 2 22 (3 2)( ( 1)(3( kk k k k k k + +...+ + = + + ++ + + == + +++ 9. Use the Principle of Mathematical Induction to prove that 2 | ( n 2 + 3 n ) for all n 1 . Ans: P (1): 2 | 1 2 + 3 1, which is true since 2 | 4. P ( k ) P ( k + 1): ( k + 1) 2 + 3( k + 1) = ( k 2 + 3 k ) + 2( k + 2), which is divisible by 2 since 2 | k 2 + 3 k and 2 | 2( k + 2). 10. Use the Principle of Mathematical Induction to prove that 2 n + 3 2 n for all n 4.
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Chapter_04 - Chapter 4 1 Suppose you wish to prove that the...

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