h4 - i , j with 1 ≤ i, j ≤ n , we have either A i ⊆ A...

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HOMEWORK 4 Question 1. Prove by induction that for all n N , n X i =1 i 2 = n ( n + 1)(2 n + 1) 6 . Question 2. Let x 6 = 1 be a real number. Prove by induction that for all n N , 1 + x + x 2 + ··· + x n - 1 + x n = x n +1 - 1 x - 1 . Question 3. Prove by induction that for all n 1, n X i =1 1 ( i + 2)( i + 3) = 1 3 - 1 n + 3 Question 4. Suppose that A 1 , . . . , A n is a list of sets such that for all
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Unformatted text preview: i , j with 1 ≤ i, j ≤ n , we have either A i ⊆ A j or A j ⊆ A i . Prove that there exists an integer k with 1 ≤ k ≤ n such that A k ⊆ A i for all i with 1 ≤ i ≤ n . ( Hint: Argue by induction on n ≥ 1.) 1...
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This note was uploaded on 01/31/2011 for the course MATH 300 taught by Professor Ctw during the Spring '08 term at Rutgers.

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