-870 (1 point) If n is a positive integer, then integer, then [8 1739]" (1 2...
FIULIONI USL NUXL Problem (1 point) If n is a positive integer, then then I-20 -361" 420 9/8 1 -1/8 (Hint: Diagonalize the matrix -20 4 -36 20 first. Note that your answer will be a formula that involves n. Be careful with parentheses.)
DEFINITION: For a positive integer n, τ(n) is the number of positive divisors of n and σ(n) is the sum of those divisors. 4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose r is a real number other than 1. Prove using mathematical induction that for every nonnegative integer n, = 1-r^n+1/1-r. 3) Prove using mathematical induction that for every nonnegative integer n, 1 + i+i! = (n+1)!. 4) Prove using mathematical induction that for every integer n>4, n!>2^n. 5) Prove using mathematical induction that for every positive integer n, 7 + 5 + 3 +.......
Let n be a positive integer. For each possible pair i, j of integers with 1 sisi<n, find an n x n matrix A with the property that 1 is an eigenvalue of A with g(1) = i and a(1) = j.
Ok = (6) Let n be a positive integer. For every integer k, define the 2 x 2 matrix cos(27k/n) - sin(2nk/n) sin(2tk/n) cos(27 k/n) (a) Prove that go = I, that ok + oe for 0 < k < l< n - 1, and that Ok = Okun for all integers k. (b) Let o = 01. Prove that ok ok for all integers k. (c) Prove that {1,0,0%,...,ON-1} is a finite abelian group of order n.
Let n be a positive integer. For each possible pair i, j of integers with 1<i<i <n, find an n xn matrix A with the property that 1 is an eigenvalue of A with g(1) = i and a(1) = j.
Write a program that reads a positive integer N, where N is greater than 1 and prints the value of the floating-point value SUM where SUM is the results of the following series. SUM = 1/1! + 2/2! + 3/3! + 4/4! + ... N/N! Note that your program should print the message “Sorry, bad N”” if N is <=1, and keep reading user input until the user enters a valid value for N. Also, note that all division operations...
2. Find a formula for Σ LVk] when n is a positive integer. 2. Find a formula for Σ LVk] when n is a positive integer.
Problem 2. Let n be a positive integer. We sample n numbers ai,...,an from the set 1, 2,...,n} uniformly at random, with replacement. Say that the picks i and j with i < j are a match if a -aj. What is the expected total number of matches? Hint: Use indicators. Wİ
Q18 12 Points For any positive integer n, let bn denote the number of n-digit positive integers whose digits are all 1 or 2, and have no two consecutive digits of 1. For example, for n - 3, 121 is one such integer, but 211 is not, since it has two consecutive 1 's at the end. Find a recursive formula for the sequence {bn}. You have to fully prove your answer.