Prove that is an integer for all n > 0.
Problem 5.1.3. Prove by induction on n that (1+ n < n for every integer n > 3.
Given H(z) as shown, determine the response y(n) function 6In-3] for all n >= 0) of the system to the discrete impulse H(z) 3z/ (z2 + 2z 2)
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
Problem 30. Prove that N, Z, Q and R are infinite sets. (HINT: Prove by induction on n that is f: NN then (3k N(Vj Nn)k> f(j). Then conclude that f cannot possibly be onto N. A similar strategy works for Z, gq and R as well.)
Please answer all!! 3. Show that if n e Z so that n is odd then 8|n2 + (n + 6)2 +6. 4. (a) Let a, b, and n be integers so that n > 2. Define: а is congruent to b mod n. The notation here is a = b (modn). (b) Is 12 = 4 (mod 2)? Explain. (c) Is 25 = 3 (mod 2)? Explain. (d) Is 27 = 13 (mod8)? Explain. (e) Find 6 integers x...
1. Let n be a positive integer with n > 1000. Prove that n is divisible by 8 if and only if the integer formed by the last three digits of n is divisible by 8.
Exercise 7. Let X be a standard normal random variable. Prove that for any integer n > 0, ELY?"] = 1207) and E[x2n+] = 0.
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
Prove: Let k be a positive integer, and set n :=2k-1(2k – 1). Then (2k+1 – 1)2 = 8n +1 Prove: Let n be a positive integer, and let s and t be integers. Show that Hire (st) = n(s) in (t) mod n.