Q6) let T(n) be a running time function defined recursively as 0, n=0 n=1 3T(n -...
1. The following function t(n) is defined recursively as: 1, n=1 t(n) = 43, n=2 -2t(n-1) + 15t(n-2), n> 3 1. Compute t(3) and t(4). [2 marks] 2. Find a general non-recursive formula for the recurrence. [5 marks] 3. Find the particular solution which satisfies the initial conditions t(1) = 1 and t(2) = 43. [5 marks] 2. Consider the following Venn diagram, illustrating the Universal Set &, and the sets A, and C. А B cat,pig mouse, horse camel...
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...
2. The Fibonacci numbers are defined recursively as follows: fo = 0, fi = 1 and fn fn-l fn-2 for all n > 2. Prove that for all non-negative integers n: fnfn+2= (fn+1)2 - (-1)" 2. The Fibonacci numbers are defined recursively as follows: fo = 0, fi = 1 and fn fn-l fn-2 for all n > 2. Prove that for all non-negative integers n: fnfn+2= (fn+1)2 - (-1)"
2t +1 if 0 <t< 2 Consider f(t) = { | 3t if t > 2. (a) Use the table of Laplace transforms directly to find the Laplace transform of f. (b) Express f in terms of the unit step function, then use Theorem 6.3.1 to find the Laplace transform of f.
The binomial coefficients C(N, k) can be defined recursively as follows: C(N,0)=1, C(N,N) = 1, and, for 0 < k < N, C(N, k) = C(N − 1, k) + C(N − 1, k − 1). Implement the following two methods inside BinomialCoefficients class, one uses recursion and the other one uses dynamic programming. 12. (10 Points) The binomial coefficients C(N, k) can be defined recursively as follows: C(N,0-1, C(N,N) = 1, and, for 0 < k < N, C(N,...
' cos(3t), t<n/2, 2. Let f(t) = sin(2t), 7/2<t< , Write f(t) in terms of the unit step e3 St. function. Then find c{f(t)}.
2. (15p) We shall consider a function A, defined by the recurrences A(0,n n+1 for n 20 for m>0 for m, n > 0 A(m, n) A(m-1, A(m, n-1)) = Observe that A(1,1) = A(0,A(1,0))=A(0,2) = 3 A(1,2 A(0, A(1, 1)) A(0,34 and it is now not hard to see (as can be proved by an easy induction) that A(1,n)n 2 for all n 20 1. (5p) Calculate A(2,0), A(2,1), A(2,2), and A(2,3) Then state (you are not required to...
all three questions please. thank you Prove that for all n N, O <In < 1. Prove by induction that for all n EN, ER EQ. Prove that in} is convergent and find its limit l. The goal of this exercise is to prove that [0, 1] nQ is not closed. Let In} be a recursive sequence defined by In+1 = -) for n > 1, and x = 1. Prove that for all ne N, 0 <In < 1....
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...
Let L:P1 --> P2 be a function defined by L(p)=tp + 1. Where p is a linear polynomial in P1. L(t+1)+L(t-1)=? 2t^2 + 2 2t^2 + 3 t^2 + t + 2 3t^2 +t+1