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Question - Math 324 Assignment 1 (due September 17)

Please remember to staple your assignment a nd to write the names of other students you
may have worked with.

1. (1.1 #6)( 5) Show that every non-empty subset X of the set of negative integers has a
greatest element.

2. (1.2 #22)( 5) Use the identity 1/(k2 − 1) = ( 1/2 ) (1/(k − 1) − 1/(k + 1)) to evaluate the
sum ∑
2 ≤ k ≤ n 1/(k2 − 1).

3. (1.3 #22)( 5) Show by mathematical induction that if h > − 1 then (1 + h)n ≥ 1 + nh for
all non-negative integers n.

4. (1.5 #38)( 5) Show that the square of every odd integer is of the form 8k + 1.

5. (3 + 3 + 4 ) Define an L-number to be a positive integer of the form 3k + ...Read More

Solution Preview - x ∈ X} is a non-empty subset so has a least element s. Then s = − g with g ∈ X and − x ≥ s = − g implies that x ≤ g for all x ∈ X. Thus g is the greatest element in X. 2. Our sum ∑ 2 ≤ k ≤ n 1/(k2 − 1) equals (1/2) ∑ 2 ≤ k ≤ n (1/(k − 1) − 1/(k + 1)) = (1/2)( ∑ 2 ≤ k

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