. Find ƒ (1), ƒ (2), ƒ (3), ƒ (4), and ƒ (5) if f (n) is defined recursively by f(0) = 3 and for n = 0, 1, 2, ... a) ƒ (n + 1) = −2f (n). b) f(n + 1) = 3f (n) + 7.

Suppose that f (n) = 0(g(n)) and f(n) = 0(h(n)), then it is ( always / sometimes / never ) the case that g(n) = 0(h(n)).

Suppose f(n) = O(F(n)) and g(n) = O(G(n)). Is it true that f(n) / g(n) = O(F(n) / G(n))? Provide a proof using the definition of big-oh, or provide values for the four functions as a counterexample.

Explain, with an example why the following definition, would not be suitable or useful: f(n) is Ω( g(n) ) if and only if there exists n0, such that:forall n ≥ n0, there exists c > 0 such that,f(n) ≥ c g(n)

Solve the following recurrences exactly:(a) T(1) = 8, and for all n ≥ 2, T(n) = 3T(n − 1) + 15.(b) T(1) = 1, and for all n ≥ 2, T(n) = 2T(n/2) + 6n − 1 (n is a power of 2)

Another recursive algorithm is applied to some data A = (a₁, ..., am) where m = 2* (i.e. 2, 4, 8,16 ...) where x is an integer ≥ 1. The running time T is characterised using the following recurrence equations: T(1) = c when the size of A is 1 T(m) = 2T (2) + c otherwise Determine the running time complexity of this algorithm.

2) Prove divisible by 3 for any integer nzo. that n(n²+s) is divisible by 3 for Let n 1 1 (1² +5) = 1(1+5)=(6/ K(K²+5) K+ 1 ((k + 1 ) ² + 5

Let f(n) y g(n) two positive asymptotic functions. Prove or disprove the following conjectures: a) f(n) = O(g(n)) implies 2f(n) = O(29(n)). b) f(n) = O(g(n)) implies g(n) = N(f(n)). c) g(n) = O((g(n))²).

In Fibonacci's model, rabbits live forever. The following modification of Definition 3.1 accounts for if n ≤ 0 if n = 1 or n = 2 G(n) = 1 (a) Compute G(n) for n = 1, 2, ..., 12. G(1) G(2) G(3) G(4) G(5) G(6) G(7) G(8) G(9) G(10) G(11) G(12) = = = = = = = = G(n-1) + G(n − 2) - G(n - 8) if n > 2. - = (b) In this modified model, how long do rabbits live? months

Prove the following by induction, substitution, or by definition that 13n+ 10 = O(n²). Definition of Big O f(n) = O(g(n)) means there are positive constants c and no, such that ≤ f(n) ≤ cg(n) for all n ≥ no.