Problem 2: Consider a sequence defined recursively as To = 1, T₁ = 2, and T₁ = Tn-1 +37-2 for n > 2.Prove that T₁ = O(2.5") and T₁ = 2(2.25").Hint: First, prove by induction that-2.25"

The main objective is to prove the given relation.

Answered: Problem 2: Consider a sequence defined…

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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4 9 16 25 1. Find the formula for an for the sequence Q ·}, assuming that the first term is 4'8' 16'32' indexed by n = 1.
9. Find the first 6 terms of the sequence {[√2n]}, n=1 10. Show that the set of all positive multiples of 3 is countable.
I. Strong Induction: Prove the following using strong induction. 2. Suppose P,, P2, P3, ... is a sequence defined by P = 5, P2 = 11, Px = 4Pg-1– 3Pg-2 for all integers k 2 3. Prove that P, = 3" + 1 for all integers n 2 1.
1. Define the sequence a, such that a₁ = 3, a2 = 5. For each n ≥ 3, Problem an = an-1 + 2an-2-2. Use strong induction to prove that for each n E N, an = 2" + 1.
4. Prove the following statements using an inductive argument: _n(n+1)(2n+1) 6 72 a. Σiª: = i=1 12 b. If a = 2a- and b = 1-(1-a)², show that a = b for all natural numbers n. Assume that a is a fixed constant. c. Let a be the sequence defined by a₁ =1, a₁ = 8, and a₁ = a₁-11 +2a2 for n ≥ 3. Use an inductive argument to prove that if b₁ =3×2"-¹ +2(−1)", the a = b for all PL 72 neN
Suppose that the sequence x0, x₁, x2... is defined by x0 = 0, x₁ = 7, and xk+2 = xk for k≥0. Find a general formula for xk. Be sure to include parentheses where necessary, e.g. to distinguish 1/(2k) from 1/2k. xk = 0
The following iterative sequence is defined for the set of positive integers: Sn/5 if n is a multiple of 5 |n-2 if n is odd Using the rule above and starting with 13, we generate the following sequence: 13 → u13 = 11 - u11 =9 ug = 7uz =5 us =1. It can be seen that this sequence (starting at 13 and finishing at 1) contains 6 terms. The below function takes as input an integer n and returns the number of terms generated by the sequence starting at n. function i-Seq(n) %3D %3D %3D ; מיט i=13; while u-1 if statement 1 u=u/5 else statement 2 end i=i+1; end statement 1 and statement 2 should be replaced by: Statement 1 is "mod(u,5)==0" and statement 2 is "u = n-2;" None of the choices Statement 1 is "n%5==0" and statement 2 is "u = u-2;" Statement 1 is "mod(u,5)==0" and statement 2 is "u = u-2;"
Suppose we have a recursive sequence ƒ1, ƒ2, ƒ3, . . .. For the purposes of this problem, it does not matter exactly how the fi are defined, only that they are recursively defined. = i=1 For integer n ≥ 1, let P(n) be the predicate that Σï-1 fi = 2fn+2 − 3. Don't worry about whether this predicate "makes sense"; we haven't defined the fi so you won't be able to "make sense" of the P(n). It's not important for this problem. Consider a proof by induction that Vn ≥ 1: P(n). Suppose we've gotten to the inductive step, and suppose that the first steps of the inductive step are k+1 k+1 Σ1 fi = f1 + f2 + Σh fi = f₁ + f2 + Σkt k+1, i=3 (fi−1 + fi−2) = f₁ + f2 + ¹ fi- 1 + k fi 2 f1 k+1 k+1 k = f1 + f2 + Σh_2 fi + [h=1² fi i=2 = f2 + Σ²²₁ fi+fi i=1 i=1 True or false: Based on the information given, we will need strong induction for this proof. True False

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Problem 2: Consider a sequence defined recursively as To= 1, T₁=2, and Tn = Tn-1 +3Tn-2 for n > 2. Prove that T O(2.5") and T (2.25"). Hint: First, prove by induction that 2.25" < T ≤ 2.5" for all n ≥ 0. =