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Data Str & Algorithm HW Solutions 11

Data Str & Algorithm HW Solutions 11 - Base case...

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11 2.22 Theorem 2.1 n i =1 (2 i ) = n 2 + n . (a) Proof : We know from Example 2.3 that the sum of the fi rst n odd numbers is n 2 . The i th even number is simply one greater than the i th odd number. Since we are adding n such numbers, the sum must be n greater, or n 2 + n . (b) Proof : Base case : n = 1 yields 2 = 1 2 + 1 , which is true. Induction Hypothesis : n 1 i =1 2 i = ( n 1) 2 + ( n 1) . Induction Step : The sum of the fi rst n even numbers is simply the sum of the fi rst n 1 even numbers plus the n th even number. n i =1 2 i = ( n 1 i =1 2 i ) + 2 n = ( n 1) 2 + ( n 1) + 2 n = ( n 2 2 n + 1) + ( n 1) + 2 n = n 2 n + 2 n = n 2 + n. Thus, by mathematical induction, n i =1 2 i = n 2 + n . 2.23 Proof
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Unformatted text preview: : Base case. For n = 1 , Fib (1) = 1 < 5 3 . For n = 2 , Fib (2) = 1 < ( 5 3 ) 2 . Thus, the formula is correct for the base case. Induction Hypothesis. For all positive integers i < n , Fib ( i ) < ( 5 3 ) i . Induction Step. Fib ( n ) = Fib ( n − 1) + Fib ( n − 2) and, by the Induction Hypothesis, Fib ( n − 1) < ( 5 3 ) n − 1 and Fib ( n − 2) < ( 5 3 ) n − 2 . So, Fib ( n ) < ( 5 3 ) n − 1 + ( 5 3 ) n − 2 < 5 3 ( 5 3 ) n − 2 + ( 5 3 ) n − 2...
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