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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 n1) 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 n16 a n2 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 m1 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.
 Spring '09
 COSTELLO
 Math, Addition, Mathematical Induction

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