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hw4sol

# hw4sol - MATH 55 Fall 2007 Prof Bernoff Harvey Mudd College...

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MATH 55 Prof. Bernoff Fall 2007 Harvey Mudd College Homework 4 – Solutions D1: Scheinerman, Section 21 (p. 168) # 3(b),(d): Prove the following by induction. (b) We wish to show for n a positive integer that 1 3 + 2 3 + · · · + n 3 = n 2 ( n + 1) 2 4 . Solution: Base case(n=1) : Note 1 2 (1 + 1) 2 4 = 4 4 = 1 = 1 3 . Induction Hypothesis : Suppose 1 3 + 2 3 + · · · + k 3 = k 2 ( k + 1) 2 4 , we wish to show 1 3 + 2 3 + · · · + ( k + 1) 3 = ( k + 1) 2 ( k + 2) 2 4 . Note that 1 3 + 2 3 + · · · + k 3 + ( k + 1) 3 = k 2 ( k + 1) 2 4 + ( k + 1) 3 (IHOP) = ( k + 1) 2 ( k 2 4 + k + 1) , = ( k + 1) 2 ( k 2 + 4 k + 4) 4 , = ( k + 1) 2 ( k + 2) 2 4 , so the formula follows by the Principle of Mathematical Induction. (d) We wish to show for n a positive integer that 1 1 · 2 + 1 2 · 3 + · · · + 1 n · ( n + 1) = 1 - 1 n + 1 . Solution: Base case(n=1) : Note 1 - 1 1 + 1 = 1 2 = 1 1 · 2 . Induction Hypothesis : Supposing 1 1 · 2 + 1 2 · 3 + · · · + 1 k · ( k + 1) = 1 - 1 k + 1 , we wish to show 1 1 · 2 + 1 2 · 3 + · · · + 1 ( k + 1) · ( k + 2) = 1 - 1 k + 2 . Note that 1 1 · 2 + 1 2 · 3 + · · · + 1 ( k + 1) · ( k + 2) = 1 - 1 k + 1 + 1 ( k + 1) · ( k + 2) (IHOP) = 1 - k + 2 ( k + 1) · ( k + 2) + 1 ( k + 1) · ( k + 2) , = 1 - k + 1 ( k + 1) · ( k + 2) , = 1 - 1 k + 2 , so the formula follows by the Principle of Mathematical Induction. 1

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2 D2: Scheinerman, Section 21 (p. 168) # 5. A group of people stand in line to purchase movie tickets. The first person in line is a woman and the last person in line is a man. Use proof by induction to show that somewhere in the line a woman is directly in front of a man. Solution: We prove the claim inductively: Base case: If the line is just those two people, then the man in back is directly behind the woman in front. Induction Hypothesis: Supposing now that the statement is true for lines up to n persons long. In a line of length n + 1, if the person directly behind the woman in front is a man, then we are done. Otherwise, the n persons behind her constitute a line in which the person in front of a woman and the person in back is a man. By the inductive hypothesis, somewhere in that sub-line we must have a woman standing in front of a man. Therefore the claim follows from the Principle of Mathematical Induction. D3: Scheinerman, Section 21 (p. 168) # 10. Prove that every positive integer can be expressed as the sum of distinct Fibonacci numbers. For example, 20 = 2 + 5 + 13 where 2 , 5 and 13 are, of course, Fibonacci numbers. Although we can write 20 = 2 + 5 + 5 + 8, this does not illustrate the result because we have used 5 twice.
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hw4sol - MATH 55 Fall 2007 Prof Bernoff Harvey Mudd College...

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