S m O | luliberty - Search X | =% Home | mylU x | MecGraw Hill Connect: MA X | Bl Class MATH 350: Discrete [l 52 Strong Inductionand = x4 = g Ay M g G %) httpsy//prod.reader-ui.prod.mheducation.com/epub/sn_f pgx i= A * 16. Prove that the first player has a winning strategy for the game of Chomp, introduced in Example 12 in Section 1.8, if the initial board is two squares wide, that is, a 2 X # board. [Hint: Use strong induction. The first move of the first player should be to chomp B e the cookie in the bottom row at the far right.] 5 17. Use strong induction to show that if a simple polygon with at least four sides is triangulated, then at least two of the triangles in the triangulation have two sides that border the exterior of the polygon. G Answer a triangulation have two sides that border the exterior of the polygon. We will prove Yn > 4P(n). The statement is true for n = 4, because there is only one diagonal, leaving two triangles with the desired property. Fix k > 4 and assume that P(j) is true for all j with 4 Aa d) f (n + 1 ) = f (n) If (n - 1 ). 5. Determine whether each of these proposed definitions is a valid recursive definition of a function ffrom the set of nonnegative integers to the set of integers. If fis well defined, find a formula for f (n) when n is a nonnegative integer and prove that your formula is valid. a) f (0) = 0, f (n) = 2f (n -2) forn 2 1 b) f (0) = 1, f (n) =f(n - 1) - 1 forn 2 1 c) f (0) = 2, f (1) = 3, f (n) = f(n - 1) -1 forn 2 2 d) f (0) = 1, f (1) = 2, f (n) = 2f (n - 2) forn 2 2 o e) f (0) = 1, f (n) = 3f (n - 1) ifn is odd and n 2 1 andf (n) = If (n - 2) if n is even and n 2 2 Answer + a) Not valid b) f(n) = 1 - n. Basis step: f(0) = 1 = 1 - 0. Inductive step: if f(k) = 1 - k, then f(k + 1) =f(k)- 1=1-k-1 =1 - (k+1). c) f(n) = 4 - n ifn > 0, and f(0) = 2. Basis step: f(0) = 2 and f(1) = 3 = 4 - 1. Inductive step (with k 2 1): f (k + 1) = f(k) - 1 = (4 -k) -1 =4- (k+1). d) f(n) = 2[(+1)/2) . Basis step: f(0) = 1 = 21(0+1)/2) and f(1) = 2 = 2[(1+1)/2). Inductive step (with k 2 1): f (k + 1) = 2f(k - 1) = 2. 2(#/21 = 2[*/21+1 = 2L((#+1)+1)/21. e) f(n) = 3" . Basis step: Trivial. Inductive step: For odd n, f(n) = 3f(n - 1) = 3 . 3"-1 = 3"; and for even n > 1, f(n) = 9f(n - 2) = 9. 3"-2 = 3" USD/EUR 0.38% Q Search 11:40 AM 7/11/2024D Q lu.liberty - Search X my Home | myLU X Lu McGraw Hill Connect: MAT X Class MATH 350: Discrete X 5.3 Recursive Definitions a X + X G https://prod.reader-ui.prod.mheducation.com/epub/sn_fdab/data-uuid-cfc57186689643d2a1a80696dee2c841 . .. ( 89 of 1018 > Aa Answer In Exercises 12-19 f, is the nth Fibonacci number. 12. Prove that f? + f2 + ... + fr = fufu+) when n is a positive integer. 13. Prove that fi + f3 + ... + fan-1 = fan when n is a positive integer. Answer Let P(n) be "fi + f3 + ... + fan-1 = fan." Basis step: P(1) is true because fi = 1 = f2. Inductive step: Assume that P(k) o is true. Then fi + f3 + ... + f2k-1 + 2k+1 = fax + f2k+1 = f2k+2 + $2(k+1). * 14. Show that fitifn-1 - fa = (-1)" when n is a positive integer. * 15. Show that fofi + fif2 + ... + fan-if2n = fun when n is a positive integer. + Answer * 16. Show that fo - fi + f2 -... - fin-1 + f2n =fan-1 - 1 when n is a positive integer. 17. Determine the number of divisions used by the Euclidean algorithm to find the greatest common divisor of the Fibonacci numbers fn and fu+1, where n is a nonnegative integer. Verify your answer using mathematical induction. Answer 18. Let NED - ENG Game score Q Search 11:41 AM 7/11/2024D Q lu.liberty - Search X my Home | myLU X Lu McGraw Hill Connect: MAT X Class MATH 350: Discrete X 5.3 Recursive Definitions a X + X G https://prod.reader-ui.prod.mheducation.com/epub/sn_fdab/data-uuid-cfc57186689643d2a1a80696dee2c841 . .. Aa a) Show that if n E S, then n = 5 (mod 10). b) Show that there exists an integer m = 5 (mod 10) that does not belong to S. Answer 28. Let S be the subset of the set of ordered pairs of integers defined recursively by Basis step: (0, 0) ES. Recursive step: If (a, b) E S, then (a + 2, b + 3) E S and (a + 3, b + 2) ES. o a) List the elements of S produced by the first five applications of the recursive definition. b) Use strong induction on the number of applications of the recursive step of the definition to show that 5 | a + b when (a, b) ES. c) Use structural induction to show that 5 | a + b when (a, b) ES. 29. Let S be the subset of the set of ordered pairs of integers defined recursively by + Basis step: (0, 0) ES. Recursive step: If (a, b) E S, then (a, b + 1) ES, (a + 1, b + 1) ES, and (a + 2, b + 1) ES. a) List the elements of S produced by the first four applications of the recursive definition. b) Use strong induction on the number of applications of the recursive step of the definition to show that a