
Proof by Mathematic Induction $(x^{n+1}-y^{n+1})=(x-y)(\sum_{i=0}^nx^{n-i}y^i)$
Which of these are linear systems? y[n] nx[n-3] y[n] x[n-1]+3
DISCRETE MATHEMATIC For question 1, Use mathematical induction to prove the statements are correct for n ∈ Z+(set of positive integers). 1. Prove that for n ≥ 1 1 + 8 + 15 + ... + (7n - 6) = [n(7n - 5)]/2 For question 2, Use a direct proof, proof by contraposition or proof by contradiction. 2. Let m, n ≥ 0 be integers. Prove that if m + n ≥ 59 then (m ≥ 30 or n ≥...
2. Prove by induction that Ση.c)-(7+1) for n > 0 and i > 0.
Suppose X, Y are independent with X ∼ N (0, 1) and Y ∼ N (0, 1). Show that the distribution of Q = X/Y follows the Cauchy distribution, i.e., f(q) = 1/π(1+q2) . Hint: Let Q = X/Y and V=Y. Find the joint pdf of Q and V and finally find the marginal pdf of Q by integrating the joint pdf of Q and V w.r.t. V: Y π(1+q2) Y V = Y . Find the joint pdf of...
part (c)
7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all x > 0. (Hint: Treat x = 0 as for x > 0 you can use L'Hospital's rule (Theorem A.11) - but remember that n is the variable, not x.) (b) Find lim - So fn(x)dx. (Hint: The answer is not 0.) (c) Why doesn't your answer to part (b) violate Proposition 7.27 Proposition 7.27. Suppose f. : G C is continuous, for n...
Strong induction
(a) For any x, y € R, if n e N then x – y is a factor of x" — y”.
2. Given x[n]— 1-ae-ja' find the DTFT of: (a) y[n] = nx[n],(b) z[n] = (n − 1)x[n] dX(92) Hint: nx[n]< > ; dΩ
Problem Show that a system with excitation x[n] and response y[n] described by yin] nx[n], is linear, time variant and static.
Want proof by induction
2.38 Show that &* () – 12v k=0
Consider the difference equation y(n) = -0.6y(n-1) - x(n-1) for n>=0 and y(-1) = 0 1. WITHOUT using Laplace or Z-Transform, determine the unit-impulse response h(n) starting from h(0).