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2+4+6+ + 2n = n(n+1) for all n ≥ 1. In your own words, what is the difference between the principle of mathematical induction proof structure and the principle of strong induction proof structure?

2+4+6+ + 2n = n(n+1) for all n ≥ 1. In your own words, what is the difference between the principle of mathematical induction proof structure and the principle of strong induction proof structure?

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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You do not need to use complete sentences; however, you must clearly indicate any assumptions you make and define any variables introduced. \[ 2 + 4 + 6 + \cdots + 2n = n(n + 1) \text{ for all } n \geq 1. \] (b) In your own words, what is the difference between the principle of mathematical induction proof structure and the principle of strong induction proof structure?
Transcribed Image Text:You do not need to use complete sentences; however, you must clearly indicate any assumptions you make and define any variables introduced. \[ 2 + 4 + 6 + \cdots + 2n = n(n + 1) \text{ for all } n \geq 1. \] (b) In your own words, what is the difference between the principle of mathematical induction proof structure and the principle of strong induction proof structure?
(a) Sketch a proof of the statement below using mathematical induction.
Transcribed Image Text:(a) Sketch a proof of the statement below using mathematical induction.
Expert Solution
Step 1

(a) Let P(n): 2+4+6+...+2n=n(n+1) .

Let n=1. 2.1=1(1+1).

So P(1) is true.

Let P(k) is true for some k≥1.

So  2+4+6+...+2k=k(k+1).

Now 2+4+6+...+2k+2(k+1)=k(k+1)+2(k+1)=(k+1)(k+2).

So P(k+1) is true when P(k) is true. But P(1) is true. Then by principle of mathematical induction we have P(n) is true for all n≥1. 

So  2+4+6+...+2n+2(n+1)=n(n+1) ∀n≥1.

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