2. Define the function a(n) as follows:a(n) =5n = 110n = 22a(n − 1) + a(n − 2) n ≥ 3and let S(n) represent a(n) < 3. Use either simple or strong induction to prove thatS(n) is true for all n ≥ 3, n E N

Given that a1=5, a2=10, an=2an-1+an-2, for n≥3 Our aim is to prove that an<3n for n≥3 using the…

Answered: 2. Define the function a(n) as follows:…

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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n - n. Use induction to prove: for any integer n ≥ 1, Σ(6j − 4) = 3n² - j=1 Base case n = 1 (6j - 4) = 2,3n² — n = 2 Inductive step Assume that for any k ≥ 1, Σ (6j − 4) = D OW! we will prove that (6j − 4) = - 1 j= Σ(6j - 4) = Σ(6j — 4) + (6-4)+ j= || j= || || = 3 + k²+ k²+ By inductive hypothesis k+ k+ -(k+1) - (k+1)
Let S(n) be a statement parameterized by a positive integer n. Consider a proof that uses strong induction to prove that for all n 2 4, S(n) is true. The base case proves that S(4), S(5), S(6), S(7), and S(8) are all true. In the inductive step, assume that for k > 8 S(j) is true for any 4
Please show the guess is correct by induction... if guess isn't right propose and prove right one.
5.2.8 More generally, use induction to prove that (a – b) | (a" – b") for any positive integers a, b, n.
5. (15) Use mathematical induction to prove that for every n E N the following is true. E2i – 2"+1 – 1 i=0 (In other notation, 20 + 21 + 22 + 23 + . .. + 2" = 27+1 .- 1)
Consider the function f: N → N given recursively by f(0) = 2 and f(n + 1) = 4 · f(n). Find f(10) * Preview
Prove that the following statement is true for all n ≥ 1 using the Principle of Mathematical Induction.
For any n > 1, let o(n) be the number of positive integers a < n with gcd(a, n) = 1. This function is called the totient function. (a) Prove that if n = pq is the product of two distinct primes, o(n) = (p − 1)(q − 1) (b) Prove that if p is prime, (p²) = p(p − 1).
Given that P(n) is the equation 1·(1!)+2· (2!) + · an integer such that n > 1, we will prove that P(n) is true for all n > 1 by induction. 6. +n· (n!) = (n + 1)! – 1, where n is (a) Base case: i. Write P(1). ii. Show that P(1) is true. In this case, this requires showing that a left-hand side is equal to a right-hand side. (b) Inductive hypothesis: Let k > 1 be a natural number. Assume P(k) is true. Write P(k). (c) Inductive step: i. Write P(k + 1). ii. Use the assumption that P(k) is true to prove that P(k+1) is true. (d) Explain why this shows that P(n) is true for all n > 1.
2. Prove the following, using induction: (a) If an+1 = an/(an +1) prove that an = ao/(nao+1) for all n ≥ 0. (b) Define fo = 0, f₁ = 1 and fn = fn-1 + fn-2 for n ≥ 2. Show that 3 fak for all k ≥ 0. (c) Show that for 2" > 4n for n > 5. (d) Show that 3 | 5 - 2" for all n. (e) Show that every n ≥ 8 can be written as 3a+5b for some a, b € N.
Exercise 2: Proof by induction Prove the following statements using induction. 1. VnEN 2" > 2n 2. Vn EN 1(2k − 1) = n² 3. For all integers n ≥ 4 it holds 2"
Let n be a natural number such that n 2 2. Use induction on n to prove each of the following. (a) If 0 1, then 3" > B.

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2. Define the function a(n) as follows: n = 1 n = 2 2a(n-1) + a(n − 2) n ≥ 3 5 a(n) = 10 and let S(n) represent a(n) < 3. Use either simple or strong induction to prove that S(n) is true for all n ≥ 3, n E N