In Fibonacci's model, rabbits live forever. The following modification of Definition 3.1 accounts fo if n ≤ 0 if n = 1 or n = 2 G(n) = G(n-1) + G(n-2) (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-8) if n > 2. (b) In this modified model, how long do rabbits live? months

This is JAVA Programming

There are m towns in a straight line, with a road joining each pair of consecutive towns. Legends say that an ordinary person in one of these towns will become a hero by completing a sequence of n quests. The first quest will be completed in their home town, but after each quest they may complete their next quest either in the same town or after moving to a neighbouring town.For example, if n = 5 and m = 4, a resident of town 2 may become a hero as follows:• begin in town 2 for quest 1,• then move to town 3 for quest 2,• stay in town 3 for quest 3,• return to town 2 for quest 4, and• move to town 1 for quest 5.Design an algorithm which runs in O(nm) time and finds the total number of waysto complete n quests.

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.

a = 0; for i in range(N): for j in reversed(range(i,N)): a = a + i + j; O(N^N) O(N*N) O(N/2) O(logN)

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

Prove by induction that fib(0)+fib(1)+…+fib(n) = fib(n+2)-1, fib(0)=0 fib(1)=1