Q1 15 Points Define the function G from naturals to naturals recursively, by the rules G(0)...
Are the following statements true or false? 1. Let a be the sequence of numbers defined by the rules a0 = 0 and, for any n, an+1 = (n + 1) - an. Then for any natural n, an is the natural denoted in Java by "n/2". 2. Let f be any function from naturals to naturals and let g(n) be the sum, for i from 1 to n, of f(i). Suppose I have a function h from naturals to...
14. (15 points) Recall that Fibonacci numbers are defined recursively as follows: fnIn-1 +In-2 (for n 2 2), with fo 0, fi-1 Show using induction that fi +f 2.+fn In+2-1. Make sure to indicate whether you are using strong or weak induction, and show all work. Any proof that does not use induction wil ree or no credit.
Q6) let T(n) be a running time function defined recursively as 0, n=0 n=1 3T(n - 1)- 2T(n - 2), n> 1 a) Find a non-recursive formula for T(n) b) Prove by induction that your answer in part (a) is correct. c) Find a tight bound for T(n).
prove by induction!
Ex 5. (15 points total] For a natural integer n > 2, define n := V1+V1+ V1 +.. n times For instance ra = V1 + V1+V1+vī. (5a) (5 points) Write ræ+1 in function of In. (5b) (10 points) Prove that for all natural integers n > 2, In & Q.
(18) Let f and g be functions from R to R that have derivatives of al orders. Let h(k) denote the kth derivative of any function. Prove using the product rule for derivatives, the fact that and induction that k +1 k=0 (19) The Fibonacci numbers are defined recursively by Fn+2 = Fn+1 Prove that the number of subsets of { 1, 2, 3, . . . , n} containing no two successive integers is E, (20) Prove that 7n...
Q2 15 Points Let n E N and A € Mnxn(R). Define trace(A) = 21=1 Qişi (i. e. the sum of the diagonal entries) and tr : Mnxn(R) +R, A H trace(A). Q2.3 5 Points Find a basis for ker(tr) and verify that it is in fact a basis. Please select file(s) Select file(s) Save Answer Q2.4 3 Points Show that for any A € Mnxn(R) there is B e ker(tr) and a € R such that A = B+a....
PROVE BY INDUCTION
Prove the following statements: (a) If bn is recursively defined by bn = bn-1 + 3 for all integers n > 1 and bo = 2, then bn = 3n + 2 for all n > 0. (b) If an is recursively defined by cn = 3Cn-1 + 1 for all integers n > 1 and Co = 0, then cn = (3” – 1)/2 for all n > 0. (c) If dn is recursively defined by...
Question 3 (15%) Function f(n) can be recursively defined as follows. f(n)- f(n -1)+4 f(n-2) f(0) 0 and f(1) = 1 (a) Write clear pseudo code to calculate f(n). (10 points
2. (6 points) (a) (3 points) The following recursively defined sequence is similar to the Fibonacci Sequence: a, = 0, Q2 = as = 1, and an+1 = an - 3an-1 + An-2 for n > 3. Calculate the 4th, 5th, and 6th terms of this sequence. (b) (3 points) Evaluate S= lim n+0 (2n? - 12n" + 161n 3n4 - 162n +1 Be careful to justify your answer by showing the rules of limits and other results that you...
Q6 10 Points This question consists of three parts. Define the normalized cardinal sine function: sin(7tx) if x # 0 πα sinc(x) 1 if x=0 Submr 36792/assignments/572431/submissions/new i) Complete the table of values for the function sinc(x) defined above. (4 points) (a is in radians) sinc(x) -3 -2.46 -2 -1.43 -1 0 1 1 1.43 2 2.46 3 ii) Plot all the points on a graph and fill in the curve passing through these points. (4 points O i W...