Problem (5), 10 points Let a0:a1, a2, be a sequence of positive integers for which ao-1, and a2n2an an+ for n 2 0. Prove that an and an+l are relatively prime for every non-negative integer n. 2n+an for n >0 Problem (5), 10 points Let a0:a1, a2, be a sequence of positive integers for which ao-1, and a2n2an an+ for n 2 0. Prove that an and an+l are relatively prime for every non-negative integer n. 2n+an for n >0
3. (14 pts.) Let the sequence an be defined by ao = -2, a1 = 38 and an = 2an-1 + 15an-2 for all integers n > 2. Prove that for every integer n > 0, an = 4(5") + 2(-3)n+1.
(1 point) 1 ak= kak-1 for all integers k a) For the following recursive sequence, find the explicit formula through iteration. ao 1 a1 = a3 a4 Explicit formula:
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
(1) Let d and m be positive integers. (a) Prove that mZ is a subgroup of dZ if and only if d divides m. (b) Let d divide m. Compute the index of mZ in dZ. (c) Compute the set of left cosets dZ/mZ.
Let ao 2 bo > 0, and consider the sequences an and bn defined by an + bn n20 (1) Compute an+l-bn+1 1n terms of Van-v/bn. (2) Prove that the sequence an is nonincreasing, that the sequence bn Is nonde- creasing, and that an 2 bn for all n 20 (3) Prove that VanVbn S Cr for all n20, where C> 0 and y>1 (give values of C and γ for which this inequality holds). Conclude that an-bn C,γ-n, where...
Solve and show work for problem 8 Problem 8. Consider the sequence defined by ao = 1, ai-3, and a',--2an-i-an-2 for n Use the generating function for this sequence to find an explicit (closed) formula for a 2. Problem 1. Let n 2 k. Prove that there are ktS(n, k) surjective functions (n]lk Problem 2. Let n 2 3. Find and prove an explicit formula for the Stirling numbers of the second kind S(n, n-2). Problem 3. Let n 2...
Let N = a,, l0" + . . . + a2 I 02 + al 10 + ao, where 0 a、 9, be the decimal expan- sion of a positive integer N. (a) Prove that 7, 11, and 13 all divide N if and only if 7, 11, and 13 divide the integer (100a2 + 10a1 +ao) - (100as + 10a4 + a3) +(100as 10a7 + a6) -... M Let N = a,, l0" + . . . + a2...
4) Let D be the set of all finite subsets of positive integers. Define a function (:2 - D as follows: For each positive integer n, f(n) =the set of positive divisors of n. Find the following f (1), f(17) and f(18). Is f one-to-one? Prove or give a counterexample.
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...