Quiz 2 Problem 7. [15 points] Every day in the life of Dangerous Dan is a potential disaster Dan may or may not spill his breakfast cereal on his computer keyboard Dan may or may not fall down the front steps on his way out the door Dan stubs his toe zero or more times Dan blurts something foolish an even number of times Let Tn be the number of different combinations of n mishaps Dan can suffer in one day For example, T3=7, because there are seven possible combinations of three mishaps spills 0 1 0 1 100 falls 00 111 0 stubs 3 2 1 blurts 000022 2 (a)Give a generating function g(a) for the sequence (To, Ti, T2,... Solution. We multiply the generating functions for spills(1+r), falls(1+a), stubs (1+x+x2+…=1/(1-x), and blurts(1+x2+x4+ (1 (1-x)(1-x2)(1-x)2 (b)Put integers in the boxes that make this equation true g(a) 1-x(1-x) Solution. -1.2 (c) Put a closed-form expression in the box that makes this equation true Remember that 1/(1-x) generates the sequence(1, 2, 3,) Solution. 2(n+1)Quiz 2 11 Problem 7. [15 points] Every day in the life of Dangerous Dan is a potential disaster: • Dan may or may not spill his breakfast cereal on his computer keyboard. • Dan may or may not fall down the front steps on his way out the door. • Dan stubs his toe zero or more times. • Dan blurts something foolish an even number of times. Let Tn be the number of different combinations of n mishaps Dan can suffer in one day. For example, T3 = 7, because there are seven possible combinations of three mishaps: spills 0 1 0 1 1 0 0 falls 0 0 1 1 0 1 0 stubs 3 2 2 1 0 0 1 blurts 0 0 0 0 2 2 2 (a) Give a generating function g(x) for the sequence {T0, T1, T2, . . .}. Solution. We multiply the generating functions for spills (1 + x), falls (1 + x), stubs (1 + x + x2 + . . . = 1/(1 − x)), and blurts (1 + x2 + x4 + . . . = 1/(1 − x2)): (1 + x)2 1 + x = 2 (1 − x)(1 − x2) (1 − x) (b) Put integers in the boxes that make this equation true: g(x) = + 1 − x (1 − x)2 Solution. −1, 2 (c) Put a closed­form expression in the box that makes this equation true: Tn = Remember that 1/(1 − x)2 generates the sequence �1, 2, 3, . . .�. Solution. 2(n + 1) − 1 = 2n + 1