MATC44 2008 - s3 - University of Toronto at Scarborough Department of Computer and Mathematical Sciences MAT C44 Winter 2008 Solutions to Assignment#3

MATC44 2008 - s3 - University of Toronto at Scarborough...

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University of Toronto at Scarborough Department of Computer and Mathematical Sciences MAT C44, Winter 2008 Solutions to Assignment #3 Problem 11. on page 261: Lucas numbers are defined by the relation l n = l n - 1 + l n - 2 , and l 0 = 2, l 1 = 1. (a) We need to show that l n = f n - 1 + f n +1 for n 1. We shall prove this by strong induction. When n = 1, l 1 = 1 and f 0 + f 2 = 0 + 1 = 1, so the equality holds. Assume that l k = f k - 1 + f k +1 for k n . Then, l n +1 = l n + l n - 1 = f n - 1 + f n +1 + f n - 2 + f n = ( f n - 1 + f n - 2 )+( f n + f n +1 ) = f n + f n +2 , by the definition of Fibonacci numbers. (b) Now, we need to show that l 2 0 + l 2 1 + · · · l 2 n = l n l n +1 +2 for n 0. Again, we prove it by induction. When n = 0, l 2 0 = 2 2 = 4, and l 0 l 1 +2 = 2 · 1+2 = 4. Assume that l 2 0 + l 2 1 + · · · l 2 n = l n l n +1 +2 for some n 0. Then, l 2 0 + l 2 1 + · · · l 2 n + l 2 n +1 = l n l n +1 +2+ l 2 n +1 = l n +1 ( l n + l n +1 )+2 = l n +1 l n +2 +2. Problem 18. on page 261: We shall compute a n with respect to the position of the first 2 in the sequence. If n = 0, then we only have the empty string, so a 0 = 1. Also, a 1 = 3. Now, if n 1, let j be the position of the first 2 in the sequence, 0 j n , and consider the three cases: 1. If j
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  • Fall '08
  • Math, Combinatorics, Fibonacci number, Scarborough, Hn, Computer and Mathematical Sciences

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