Donnie Smith Assignment Unit15 Recursion due 09/20/2017 at 11:59pm MST Mohacsy MAT 243 ONLINE A Fall 2017 1. (1 point) Write down the first five terms of the following recursively defined sequence. (incorrect) 4. (1 point) For the sequence an = an1 + an2 and a1 = 4, a2 = 5, its first term is ; ; its second term is its third term is ; ; its fourth term is its fifth term is . a1 = 1; an+1 = 2an 1 , , , , Answer(s) submitted: Answer(s) submitted: (incorrect) 2. (1 point) Find f(1), f(2), f(3) and f(4) if f(n) is defined recursively by f (0) = 4 and for n = 0, 1, 2, . . . by: (a) f (n + 1) = 2 f (n) f (2) = f (3) = f (4) = f (1) = (incorrect) (b) f (n + 1) = 3 f (n) + 4 f (1) = f (2) = f (3) = f (4) = (b) f (n + 1) = f (n)2 3 f (n) 3 f (1) = f (2) = f (3) = f (4) = 5. (1 point) Give a recursive definition of the following sequences {an }, n = 1, 2, 3, ..... Put the appropriate letter next to the corresponding sequence. 1. 2. 3. 4. Answer(s) submitted: a) b) c) d) an = 6 an = 6n an = 6n an = 6n + 1 an+1 = an + 6 for n 1 and a1 = 6 an+1 = an for n 1 and a1 = 6 an+1 = an + 6 for n 1 and a1 = 7 an+1 = 6an for n 1 and a1 = 6 Answer(s) submitted: (incorrect) (incorrect) 3. (1 point) For the sequence an = 2(an1 2) and a1 = 2, its first term is ; its second term is ; its third term is ; its fourth term is ; its fifth term is . 6. (1 point) Match each of the given sets with the correct recursive definition. 1. S = {x Z | x is even} 2. S = {2n | n Z+ } 3. S = {x Z+ | x is even} Answer(s) submitted: 1 a) 2 S, and if x S and y S then x + y S b) (1, 2) S, (2, 1) S and if (a, b) S then (a + 3, b) S, (a, b + 3) S, (a + 1, b + 2) S and (a + 2, b + 1) S b) 2 S, and if x S then 2x S c) 2 S, and if x S and y S then x y S c) (1, 1) S and if (a, a) S then (a + 1, a + 1) S and if (a, b) S, then (a, b + a) S Answer(s) submitted: d) 2 S, and if x S then 2 x S Answer(s) submitted: (incorrect) 8. (1 point) Consider the following inductive definition of a version of Ackermann's function: 2n if m = 0 0 if m 1 and n = 0 A(m, n) = 2 if m 1 and n = 1 A(m 1, A(m, n 1)) if m 1 and n 2 Find the following values of the Ackermann's function: A(2, 2) = A(1, 2) = A(1, 3) = A(3, 1) = A(1, 1) = A(3, 3) = (incorrect) 7. (1 point) Give a recursive definition of each of these sets of ordered pairs of positive integers. (Hint: plot the points in the set in the plane and look for lines containing points in the set. 1. S = {(a, b) | a Z+ , b Z+ , and a + b is odd} 2. S = {(a, b) | a Z+ , b Z+ , and 3 | a + b} 3. S = {(a, b) | a Z+ , b Z+ , and a | b} Answer(s) submitted: a) (1, 2) S, (2, 1) S and if (a, b) S then (a + 2, b) S, (a, b + 2) S and (a + 1, b + 1) S (incorrect) c Generated by WeBWorK, http://webwork.maa.org, Mathematical Association of America 2