Let ao 2 bo > 0, and consider the sequences an and bn defined by an...
4. Find the first six terms of the sequences defined by each of these recursive definitions: (a) ao = 1; a1 = 1; an = An-1 + 2an-2 for n>2 (b) bo = 1; bı = 2; bn = 2bn-1-bn-2 for n> 2 (c) co =1, cı=2; Cn Cn-2-0-1 for n> 2
8. Consider the following simultaneous homogeneous recurrence relations: 3a-12bn-1 bn-an-1 + 2bn-1 for n > 1, with initial conditions ao 1 and bo - 0 (a) Find the generating function for an and then solve for an b) What is the homogeneous recurrence relation that an satisfies? (c) Repeat (a) and (b) for bn 72. 8. Consider the following simultaneous homogeneous recurrence relations: 3a-12bn-1 bn-an-1 + 2bn-1 for n > 1, with initial conditions ao 1 and bo - 0...
Let e-Σ (Application of Cauchy product) for x e R. Exercise 21: n-0 a) Show that bk for all b) Let (bn)neNo be the recursion defined by bo - 1 and bn- k-0 n E N. Show that bn-- Hint: Use a) with e*e*1 and the inverse of a power series found in the lecture. Let e-Σ (Application of Cauchy product) for x e R. Exercise 21: n-0 a) Show that bk for all b) Let (bn)neNo be the recursion...
Discrete Mathematics 3. The sequence bo, bi, b2, is defined as follows: bo 0, bnd for integers n 22, bn- ehne (a) Calculate b2, b3, ba and bs (b) Use part (a) to guess a formula for bn for all integers n 20. c) Prove by induction on n that your guess in part (b) is correct.
PROVE BY INDUCTION Prove the following statements: (a) If bn is recursively defined by bn = bn-1 + 3 for all integers n > 1 and bo = 2, then bn = 3n + 2 for all n > 0. (b) If an is recursively defined by cn = 3Cn-1 + 1 for all integers n > 1 and Co = 0, then cn = (3” – 1)/2 for all n > 0. (c) If dn is recursively defined by...
(8 marks) Suppose that bo, bi,b2,... is a sequence defined as follows: bo 1, b 2, b2 3, and b bk-1 + 4bk-2 +5bk-3 for all integers k 2 3. Prove by mathematical induction that bn S 3" for all integers n 2 0.
Prove the following Definition 6.6.1 (Subsequences). Let (an) =) and (bn), m=0 be sequences of real numbers. We say that (bn)is a subsequence of (an) a=iff there exists a function f :N + N which is strictly increasing (i.e., f(n + 1) > f(n) for all n EN) such that bn = f(n) for all n E N.
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.
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...
Q6 6 Points Let an and bn be 2 convergent sequences. Let A = limno an and B = limno br. Prove that limno anbn AB = You may use the following inequality anbn - AB B) + Blan - A), exercise 4, triangular inequalities, among other results proved in class or above. : an (bn