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Unformatted text preview: brieﬂy. 8. Let x ∈ Z . Prove the following: 2 x 2 + x + 1 is even iﬀ x is odd. 9. Let m,n ∈ Z . Show that there doesn’t exist an integer k such that (3 m +2)(3 n +2) = 3 k +2. 10. Let x ∈ R . Prove the following: If  x1  < 1 then  x 24 x + 3  < 3 11. Let h a n  n ∈ N i be the sequence deﬁned recursively as follows: a 1 = 1 a n +1 = a n + 3 n ( n + 1) for n ∈ N Prove that a n = n 3n + 1 for all n ∈ N . 12. Show that ∀ n ∈ N the following holds: n X k =1 2 k = 2 n +12 13. Recall that the Fibonacci Numbers are deﬁned by f = 0, f 1 = 1 and f n = f n1 + f n2 for all n > 1. Show that for every natural number n the following holds: 2 n1 f + 2 n2 f 1 + ... + 2 f n1 = 2 nf n +2...
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 Spring '07
 GHEORGHICIUC
 Math, Natural Numbers, Natural number, Fibonacci number

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