MAS111_09_assignment_01 - 7. Prove that for any positive...

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NANYANG TECHNOLOGICAL UNIVERSITY MAS 111 FOUNDATION OF MATHEMATICS ASSIGNMENT 1 TUTORIAL TIME: 17/08/09 1. Prove that the square of any odd integer leaves remainder 1 upon di- vision by 8. 2. Prove that the product of two numbers of the form 4 k +3 is of the form 4 k + 1. 3. Prove that for natural numbers m,n , m 2 + n 2 is a multiple of 3 if and only if both m and n are multiples of 3. 4. Prove that there are infinitely many n such that n 2 + 23 is divisible by 24. 5. (a) Prove that the square of any integer is of the form 4 k or 4 k + 1. (b) Prove that no integer in the sequence 11 , 111 , 1111 , 11111 , · · · is the square of an integer. 6. Prove that 11 n +2 +12 2 n +1 is divisible by 133 for all natural numbers n .
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Unformatted text preview: 7. Prove that for any positive natural number n , 1 · 2 + 2 · 5 + · · · + n · (3 n-1) = n 2 ( n + 1) . 8. Prove that for any positive natural number n , 1 2 + 3 2 + · · · + (2 n-1) 2 = n (4 n 2-1) 3 . 9. Prove that for any positive natural number n , 1 n + 1 + 1 n + 2 + · · · + 1 3 n + 1 > 1 . Note that for different n , the numbers of terms on the left-hand side are different. This inequality can be applied to show that 1 1 + 1 2 + · · · + 1 n + · · · does not converge, which will be studied in CALCULUS III. 10. Prove that for all natural numbers n > 1, 4 n n + 1 < (2 n )! ( n !) 2 ....
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MAS111_09_assignment_01 - 7. Prove that for any positive...

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