(2) Let an be the sequence defined by:a₁ = 1a2 = 2an an-1 + 2an-2 for n ≥ 3.(a) Find a3,..(b) Use induction to prove that an > 0 for all n € N.an(c) Use induction to prove thatan+1(d) Use (b) and (c) to conclude that an < an+1 for all n € N..... ag.< 1 for all n € N.

Given that a1=1 , a2=2 and an=an-1+2an-2 for n≥3

Answered: (2) Let an be the sequence defined by:…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Use iteration to find a closed formula for the sequences below. Simplify your answer whenever possible. a.) Let (a_n)be the sequence defined by the recursive relation an=3a_(n−1)+4, with initial term a0=3. b.) Let (a_n) be the sequence defined by the recursive relation an=a_(n−1)+6, with initial term a0=5 c.) Let (a_n)be the sequence defined by the recursive relation an=a_(n−1)+5n+1 with initial term a0=0. d.) Let (a_n)be the sequence defined by the recursive relation an=(a_(n−1)) / (1+a_(n−1)) with initial term a0=1.
You don't need part 1a, they are just saying it's a sequence.
8. Let fa,) be a sequence defined recursively by a, = 2 and for all n2 2, a,= a-1+ 2n. Prove via induction that a, = n? + n for all natural numbers n. 6
1. а) Suppose that fis defined recursively by: f (0) = 5 and f(n+1) = 2 fn +5. Find A1), A2), (3) and f4)? b) For which positive integer n is it true that 2">n³? c) Prove your answer in (b) above using mathematical induction. d) Give a recursive definition of the sequence {an}, n= 1, 2, 3... if an = 2" + 1 e) Use your definition in (d) above to find a10, and a15 f) Let A = {1, 2, {{1,2}}}. Find the power set P(A)
c. Let a be the sequence defined by a₁ = 1, a₁ = 8, and a = a₁ +2a2 for n ≥ 3. Use an inductive argument to prove that if b = 3x2¹ +2(-1)", the a = b for all " neN
(4) Suppose that a, b are positive integers such that 62 – 2a? = 1. Prove that there exist positive integers c, d such that and are consecutive terms in some Farey sequence and a 1 V2 (Hint: guess expressions for c, d in terms of a, b that seem to work, then prove that they do.) b d'
(1) Using the appropriate formula, find the 17th term of the sequence 7,4, 1,..... (2) If h is defined by h(x) = |x| + 2, (a) What is the range of h if the domain of h is the interval [-8,2)? (b) What is the range of h if the domain of h is the set of negative integers?

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(2) Let an be the sequence defined by: a₁ = 1 a2 = 2 an an-1 + 2an-2 for n ≥ 3. (a) Find a3,.. (b) Use induction to prove that an > 0 for all n € N. an (c) Use induction to prove that an+1 (d) Use (b) and (c) to conclude that an < an+1 for all n € N. .... ag. < 1 for all n € N.