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Prove that 2" < n! for any integern 2 4

Prove that 2" < n! for any integern 2 4

Advanced Engineering Mathematics
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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I am working on the proofs for Questions 1 and 2 and was looking for some assistance on how to prove these questions.

1) Prove that 2" < n! for any integer n > 4 2) Prove that 4 divides 7" - 3" for eachn EN 3) Prove that sin nx < n| sin x| for all n EN and xE R. 4) Define f : N → R as follows: 3 f(1) = 3, f(2) = and for n > 3 f(n) = f(n- 1) + f(n - 2) %3D 2' 2. Prove that f(n) = 2+ (-)"-1 for all n EN 1 1 · . ·+ 4 1 TS2 Vn E N 5) Prove that 1+-+-+ 2n-1 x +y 6) Let x > 0 and y > 0, prove that Ty < that ry S
Transcribed Image Text:1) Prove that 2" < n! for any integer n > 4 2) Prove that 4 divides 7" - 3" for eachn EN 3) Prove that sin nx < n| sin x| for all n EN and xE R. 4) Define f : N → R as follows: 3 f(1) = 3, f(2) = and for n > 3 f(n) = f(n- 1) + f(n - 2) %3D 2' 2. Prove that f(n) = 2+ (-)"-1 for all n EN 1 1 · . ·+ 4 1 TS2 Vn E N 5) Prove that 1+-+-+ 2n-1 x +y 6) Let x > 0 and y > 0, prove that Ty < that ry S
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Step 1

We’ll answer the first question since the exact one wasn’t specified. Please submit a new question specifying the one you’d like answered.

(1) We have to prove that 2n<n! for any integer n≥4

We will prove by principal of mathematical induction.

Let n=4

⇒24<4!⇒14<24

Hence, inequality is true for n=4

Let n=5

⇒25<5!⇒32<120

Hence, inequality is true for n=5

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