Can you explain this? 11:08AAA study.240tutoring.com(L40+ Return to courseA proposition states that "For all integers nsuch that n > 4, then n2 < 2"." A studentproves this proposition using mathematicalinduction. They have shown the base case,and are on the inductive step, which isdisplayed below:Suppose now as inductive hypothesisthat k2 < 2k for some k > 4. Then2k+1 = 2 x 2k > 2k². But 2k² >(k + 1)? + 2k2 > k² + 2k +1 +k2 > 2k + 1 and since k > 4, k² >4k > 2k + 2 > 2k + 1 so that k2 >2k + 1. Hence 2k² > (k + 1)².Why was the student able to assume that2 x 2k > 2k²?ВackNextIncorrect

Answered: A proposition states that "For all…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

100%

Can you explain this?

11:08
AA
A study.240tutoring.com
(L40
+ Return to course
A proposition states that "For all integers n
such that n > 4, then n2 < 2"." A student
proves this proposition using mathematical
induction. They have shown the base case,
and are on the inductive step, which is
displayed below:
Suppose now as inductive hypothesis
that k2 < 2k for some k > 4. Then
2k+1 = 2 x 2k > 2k². But 2k² >
(k + 1)? + 2k2 > k² + 2k +1 +
k2 > 2k + 1 and since k > 4, k² >
4k > 2k + 2 > 2k + 1 so that k2 >
2k + 1. Hence 2k² > (k + 1)².
Why was the student able to assume that
2 x 2k > 2k²?
Вack
Next
Incorrect

Expert Solution

Step 1

An inductive hypothesis is part of an argument by mathematical induction that some predicate $P$ holds for all natural numbers $N .$

It's actually more general than that since mathematical induction can be used on any structure defined inductively, but for this answer, let's stick to simple induction on $N .$

$N,$ the natural numbers, is defined inductively by saying it has an initial element 0 (some people prefer to start with 1; that's fine, too), and for each element $n,$ there is another element $σ (n)$ called the successor of n, which, after addition is defined, can be identified with $n + 1.$