AP 06 - Jan 30 - n . 8. Show that (1 3 + 2 3 + + n 3 ) 3(1...

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1 Math 2602 M1/M2 Spring 2007 Additional Problems 6 Telescobing + Divisibility 1 Thursday, Jan 30th 1. Show that n X k =1 k !( k 2 + k + 1) = ( n + 1)!( n + 1) - 1 . Hint: k 2 + k + 1 = ( k + 1) 2 - k. Prove your answer by mathematical induction. 2. Let a n be defined recursively by a n = a b n/ 3 c + a b ( n +1) / 3 c + a b ( n +2) / 3 c - 4 , n 2 , a 0 = 2 , a 1 = 7 . Use the Principle of Mathematical Induction to show that a n = 5 n + 2 for n 0 . Hint: any integer number can be in the form of either 3 k or 3 k + 1 or 3 k + 2 . 3. Let A n be defined recursively by A n = n + A b n/ 2 c + A b n/ 5 c , n 1 , A 0 = 0 . Use the Principle of Mathematical Induction to show that A n < 10 n for n 1 . 4. Do there exists positive integers x, y, z such that 21 x - 9 y + 15 z = 2602 ? If yes, find such. If no, explain why. 5. Prove by using the mathematical induction that the positive integer 8 n + 6 is divisible by 7 for any integer number n 1 . 6. Prove by using the mathematical induction that the product n ( n + 2)( n + 4) is divisible by 3 for any integer number n 0 . 7. Prove by using the mathematical induction that the positive integer 5 n + 6 · 7 n + 1 is divisible by 8 for any integer number
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Unformatted text preview: n . 8. Show that (1 3 + 2 3 + + n 3 ) 3(1 5 + 2 5 + + n 5 ) , n 1 . Hint: Use the Principle of Mathematical Induction to show that for n 1 1 3 + 2 3 + + n 3 = n 2 ( n + 1) 2 4 1 5 + 2 5 + + n 5 = n 2 ( n + 1) 2 (2 n 2 + 2 n-1) 12 1 T. Yolcu, School of Mathematics at Georgia Tech. Email: tyolcu@math.gatech.edu . 2 9. Let a n be dened recursively by a n = 5 a n-1-6 a n-2 n 3 , a 1 = 5 , a 2 = 13 . Use the Principle of Mathematical Induction to show that a n = 2 n + 3 n for n 1 . 10. Fibonacci sequence F n is dened by F n +2 = F n +1 + F n with F = 0 , and F 1 = 1 . Use induction to prove the following statements: (a) For n , m 1 , F n + m = F m F n +1 + F m-1 F n . (b) If n m, then F n F m (c) For n 1 , m 1 , F gcd ( m,n ) = gcd ( F n , F m ) ....
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This note was uploaded on 05/12/2010 for the course MATH 2602 taught by Professor Costello during the Spring '09 term at Georgia Institute of Technology.

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AP 06 - Jan 30 - n . 8. Show that (1 3 + 2 3 + + n 3 ) 3(1...

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