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Unformatted text preview: ,a 3 ,... as follows: a 1 = 3 a 2 = 6 a n = 5 a n16 a n2 + 2 , for n ≥ 3 Prove that a n = 1 + 2 n1 + 3 n1 for all n ∈ N . EXERCISES: You do not need to submit these questions but you should make sure that you are able to answer them; 1. Spivak # 1.3(i)(iv); use P1P9 for the proofs. 1.4(xiii)(xiv), 1.5(i)(iv); use P1P12. 1.11(iii)(v), 1.12(i)(v)(vi)(vii), 1.20, 2.1(i)(ii), 2.5(a). 2. Prove that (1) x =x for all x ∈ R . Use P1P12. 3. Factor x 23 x + 2. Use P1P12. 4. Show each of the following: (a) If  x  ≤ 2 then  x 3x 2cos ( x )  ≤ 13. (b) If  x  ≥ 1 then  x  3 ≥  x  . (c) If  x  ≤ 1 2 then  x 1x 2  ≤ 2 3 . 5. Suppose that x 1 ,x 2 ,...,x n are real numbers. Prove that  x 1 + x 2 + ··· + x n  ≤  x 1  +  x 2  + ··· +  x n  . 2...
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 Summer '11
 Katherine
 Math, Englishlanguage films, Following, Zagreb, Computer & Mathematical Sciences

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