# Can anyone explain an interesting induction result I got

• cnwilson2010
In summary, the conversation is about proving the inequality n2<=2n for all natural numbers n using induction. The attempt at a solution involved trying values for n and noticing that n=3 did not work. The summary also includes a hint for the proof.
cnwilson2010

## Homework Statement

n2<=2n
n is a natural number

For what values of n is the statement true and prove by induction.

## The Attempt at a Solution

I tried 1 and it worked, I tried 2 and it worked, just for fun I tried 3 and it didn't work, so I assumed the opposite and went to town but could never get the (k+1) portion to make any sense, so after hours and hours, I tried n=4 and it worked, then n=5 works, etc. Why doesn't n=3 work and all the others do? And how do you phrase this in proof language?

cnwilson2010 said:

## Homework Statement

n2<=2n
n is a natural number

For what values of n is the statement true and prove by induction.

## The Attempt at a Solution

I tried 1 and it worked, I tried 2 and it worked, just for fun I tried 3 and it didn't work, so I assumed the opposite and went to town but could never get the (k+1) portion to make any sense, so after hours and hours, I tried n=4 and it worked, then n=5 works, etc. Why doesn't n=3 work and all the others do? And how do you phrase this in proof language?
It looks like the statement to prove is this:
Show that for any integer n, where n >= 4, that n2 <= 2n

Your base case would need to be at least 4.

As to why this inequality isn't true for n = 3, look at the graphs of y = x2 and y = 2x, for x >= 0. The two graphs intersect at (1, 1) and (2, 4), and (4, 16). Between x = 2 and x = 3, the graph of the quadratic is above the graph of the exponential. After the graphs cross again at (4, 16), the exponential grows more steeply than the quadratic, and the two never cross again. That's essentially what you're proving by induction.

2. Assume that the statement is true for n = k.
3. Show that the statement being true for n = k implies that the statement is also true for n = k + 1.

The proof by induction of this inequality is quite fun and I suggest that you play with it some more. I will also provide the small hint that at some point you will need to show that $2k+1<k^{2}$ at some point.

## 1. What is an induction result?

An induction result is an outcome or conclusion that has been derived through the process of induction, which is a method of reasoning where specific observations are used to make generalizations or predictions about a larger set of data.

## 2. How is induction used in scientific research?

Induction is commonly used in scientific research to form hypotheses and theories based on observations and experiments. It allows researchers to make predictions and draw conclusions from a limited set of data.

## 3. Can you give an example of an interesting induction result?

An example of an interesting induction result could be the discovery of the law of gravity by Sir Isaac Newton, who observed the movements of objects and used induction to formulate a theory that explained the force of gravity.

## 4. What makes an induction result valid?

To be considered valid, an induction result must be supported by a sufficient amount of evidence and must be able to make accurate predictions about future observations. It is also important for the process of induction to be logical and free from biases or errors.

## 5. How can I explain an interesting induction result to others?

To explain an interesting induction result to others, it is important to provide a clear and concise explanation of the process of induction and how it led to the result. It may also be helpful to provide examples or visual aids to make the explanation more accessible and understandable.

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