sierra - a 1 = 3 and let a n = 2 a n-1-1 for all n ≥ 2[1...

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Math 210 Section 011: Test #3 NAME: This test has 6 pages . The points per page are 4, 4, 5, 5, 5, 5. [2 points] Question 1a. Find all solutions of the congruence 2 x 18 (mod 50), where 0 x < 50. [2 points] Question 1b. Find all solutions of the congruence 5 x 1 (mod 11), where 0 x < 11. 1
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[4 points] Question 2. Find all solutions of the system of congruences 2 x + 3 y 0 (mod 7) 3 x + 5 y 6 (mod 7) where 0 x < 7 and 0 y < 7. 2
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[5 points] Question 3. Find all solutions of the system of congruences x 3 (mod 8) x 7 (mod 9) where 0 x < 72. 3
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[5 points] Question 4. Prove that n 3 + 2 n is divisible by 3 for all natural numbers n . (You may give either an inductive or non-inductive proof, as long as your proof is correct.) 4
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Question 5. Consider the following recursively defined sequence. Let
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Unformatted text preview: a 1 = 3, and let a n = 2 a n-1-1 for all n ≥ 2. [1 point] (i) Give the values of a 2 , a 3 , a 4 , a 5 , and a 6 . [2 points] (ii) Guess a non-recursive formula for a n . [2 points] (iii) Use induction to prove that your formula from (ii) is correct. 5 [5 points] Question 6. Use induction to prove that n ! > n 3 for all n ≥ 6. (Recall that n ! means 1 × 2 × 3 × ··· × n .) To save you time with computations, I include some values of n ! and n 3 . Note that there’s a very good reason I told you to start with n = 6. n 1 2 3 4 5 6 n ! 1 2 6 24 120 720 n 3 1 8 27 64 125 216 6...
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