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1. The following function t(n) is defined recursively as: 1, n=1 t(n) = 43, n=2 -2t(n-1)...
(1 point) Find the first six terms of the recursively defined sequence 251/2 n-1 Sn = for n > 1, and s1 = 1. 4. first six terms = (Enter your answer as a comma-separated list.)
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).
[D] (8pts) Consider the recursively defined function below. F(1)=2, F(2) = 1, and F(n) = F(n-1) +2F(n-2) for n > 3. Find the value of F(3), F(4) and F(5). Do any necessary work in the space below and write your answers in the blanks provided. Answers: F(3) = - F(4) = — F(5) = -
Suppose that the following subset T of binary strings is defined recursively: • Basis: 1 is in T • Recursively, if the binary string s is in T, then so are the strings Os, so, 181, 11s and s11 1. Carefully show why the string 011001 must be in the set T. 2. Provide an argument that shows that if s is a string in T of length n and s has an odd number of 1s, then all strings...
A periodic function ft) of period T-2 is defined as ft)-2t over the period (a) Sketch the function over the interval -3m<<3x. [3] (b) Find the cireular frequency a and the symmetry of the function (odd, even or neither). 21 (e) Determine the trigonometric Fourier coefficients for the function f) [10] (d) Write down its Fourier series for n=0, 1, 2, 3 where n is the harmonic number. [5] (e) Determine the Fourier series for the function g(t)-2r-1 over the...
Given the sequence an defined recursively as follows: an 3an-1+2 for n 2 1 Al Terms of a Sequence (5 marks) Calculate ai , аг, аз, а4, а5 Keep your intermediate answers as you will need them in the next question. A2 Iteration (5 marks) Using iteration, solve the recurrence relation when n21 (i.e. find an analytic formula for an). Simplify your answer as much as possible, showing your work and quoting any formula or rule that you use. In...
1. Let T : P (R) Pn+1(R) be defined: T(p()) = (x + 1)p(x + 2) (a) (2 marks) Show that T is a linear transformation. (b) (3 marks) Is T one-to-one? Describe ker(T). What is the rank of T? (c) (8 marks) Find a matrix representation for T with respect to the standard bases {1, 2, ..., 2"} for Pn and {1, 2, ..., xn+1} for Pn+1 if n = 4. (d) (5 marks) Let D : Pn+1(R) +...
3. The linear transformation T : Q3 → Q2 is defined T-1 (a) Find an expression for T b [4 marks] (b) Find Mats.c(T) where and C- [2 marks]
Applied Mathematics Laplace Transforms 1. Consider a smooth function f(t) defined on 0 t<o, with Laplace transform F(s) (a) Prove the First Shift Theorem, which states that Lfeatf(t)) = F(s-a), where a is a constant. Use the First Shift Theorem to find the inverse trans- form of s2 -6s 12 6 marks (b) Prove the Second Shift Theorem, which states that L{f(t-a)H(t-a))-e-as F(s), where H is the Heaviside step function and a is a positive constant. Use the First and...
(24%) Find the Fourier transform of the signals given below: 1) x() 4e U() x(t) = e-3,Cos(12m)U(t) 2) x(t) = 36(1 + 4) + 26(1) + 43(1-5) 3) x[n]=(0.7)"U(n) 4) (24%) Find the Fourier transform of the signals given below: 1) x() 4e U() x(t) = e-3,Cos(12m)U(t) 2) x(t) = 36(1 + 4) + 26(1) + 43(1-5) 3) x[n]=(0.7)"U(n) 4)