Question

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, (n) = f(n-3) for n ≥ 3.

c) f(0) = 0, f(1) = 1, f(n) = 2 f(n + 1) for n ≥ 2

d) f(0) = 0, f(1) = 1, f(n) = 2 f(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.

Answer 1