AP 02 - Jan 18

# AP 02 - Jan 18 - F n = 1 5 " 1 + 5 2 ! n- 1- 5 2 ! n #...

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Math 2602 M1/M2 Spring 2007 Additional Problems 2 The Principle of Mathematical Induction 1 Thursday, Jan 18th 1. Among the many interesting sequences of numbers encountered in discrete mathematics, one ﬁnds harmonic numbers h 1 , h 2 , h 3 , . . . , where h n = 1 + 1 2 + 1 3 + ··· + 1 n , n 1 . Recursively, h n +1 = h n + 1 n + 1 , n 1 . Use the Principle of Mathematical Induction to show that for n 1 ( a ) h 1 + h 2 + h 3 + ··· + h n = ( n + 1) h n - n ( b ) 1 + n 2 h 2 n ( c ) h 1 + 2 h 2 + 3 h 3 + ··· + nh n = n ( n + 1) 2 h n - n ( n - 1) 4 . 2. Use the Principle of Mathematical Induction to show that (cos θ + i sin θ ) n = cos( ) + i sin( ) , n 1 . ( i 2 = - 1) 3. We build an exponential tower 2 2 2 2 . . . by deﬁning a 0 = 1 and a n +1 = ( 2) a n for n N . Use the Principle of Mathematical Induction to show that a n is monotonically increasing and bounded above by 2. 4. Use the Principle of Mathematical Induction to show that any integer n 14 can be written as a sum of 3’s and/or 8’s. ( E.g. 14 = 3+3+8 , 15 = 3+3+3+3+3 , 16 = 8+8) 5. Fibonacci sequence F n is deﬁned by F n +2 = F n +1 + F n with F 0 = 0 , and F 1 = 1 . Use the Principle of Mathematical Induction to show that

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Unformatted text preview: F n = 1 5 " 1 + 5 2 ! n- 1- 5 2 ! n # , n . n X k =1 F 2 k = F n F n +1 F 1 F 2 + F 2 F 3 + + F 2 n-1 F 2 n = F 2 2 n 1 T. Yolcu, School of Mathematics at Georgia Tech. Email: tyolcu@math.gatech.edu . 1 6. Let f ( x ) = x 2-1 . Dene f n ( x ) = f ( f ( ( f ( f | {z } n times ( x ))) )) , n 1 . Namely, f n +1 = f ( f n ( x )) = f n ( f ( x )) for n 1 . Use the Principle of Mathematical Induction to show that (a) f n ( x ) = a, where a <-1 , has no solution. (b) f n ( x ) = a, where a > , has only two distinct solutions. Hint: Use f n +1 ( x ) = f ( f n ( x )) = f 2 n-1 and part (a). (c) f n ( x ) = a, where a = 0 , has n + 1 distinct solutions. Hint: Use f n +1 ( x ) = f 2 ( f n-1 ( x )) = f 2 n-1 ( f 2 n-1-2) and part (b). 2...
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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 02 - Jan 18 - F n = 1 5 " 1 + 5 2 ! n- 1- 5 2 ! n #...

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