Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. if f is well defined, find a formula for f (n) when n is a nonnegative integer and prove that your formula is valid. a) f (0) = 1, f (n) = −f (n − 1) for n ≥ 1 b) f (0) = 1, f (1) = 0, f (2) = 2, f (n) = 2f (n − 3) for n ≥ 3 c) f (0) = 0, f (1) = 1, f (n) = 2f (n + 1) for n ≥ 2 d) f (0) = 0, f (1) = 1, f (n) = 2f (n − 1) for n ≥ 1 e) f (0) = 2, f (n) = f (n − 1)if n is odd and n ≥ 1 and f (n) = 2f (n − 2) if n ≥ 2

In a test for esp (extrasensory perception), the experimenter looks at cards that are hidden from the subject. each card contains either a star, a circle, a wave, a cross or a square.(five shapes) as the experimenter looks at each of 20 cards in turn, the subject names the shape on the card. when the esp study described above discovers a subject whose performance appears to be better than guessing, the study continues at greater length. the experimenter looks at many cards bearing one of five shapes (star, square, circle, wave, and cross) in an order determined by random numbers. the subject cannot see the experimenter as he looks at each card in turn, in order to avoid any possible nonverbal clues. the answers of a subject who does not have esp should be independent observations, each with probability 1/5 of success. we record 1000 attempts. which of the following assumptions must be met in order to solve this problem? it's reasonable to assume normality 0.8(1000), 0.2(1000)%30 approximately normal 0.8(1000), 0.2(1000)% 10 approximately normal srs it is reasonable to assume the total number of cards is over 10,000 it is reasonable to assume the total number of cards is over 1000