Prove 1-n^2>0: 3n-2 is Even Integer

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In summary, the conversation discusses a problem where n must be proven to be an even integer given the condition that 1-n^2>0. The speaker presents their solution, which involves setting n=0 and showing that 3n-2 is even. However, the statement of the problem is questioned for its validity and simplicity. The speaker also expresses their difficulty in wording the proof accurately.
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bonfire09
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Let nεZ,Prove that 1-n^2>0, then 3n-2 is an even integer.

I proved it like this. I think its right but I am not able to word it correctly.

Since 1-n^2>0 therefore n=0. Then 3n-6=3(0)-2=-2. Since 0 is an integer, 3n-6 is even.

How can I learn to word this correctly because I am having some trouble with it?
 
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Are you sure you have the statement of the problem correct? It looks crazy. As you say 1 - n2 > 0 would imply n = 0, which makes the rest of it much too easy. Also, why would anyone ask you to prove 3n-2 is an even integer when n is already known to be an integer? Why not just ask you to show n is even?
 

1. How do you prove the statement 1-n^2>0: 3n-2 is Even Integer?

To prove this statement, we can use the method of mathematical induction. First, we can show that the statement is true for the base case of n=1. Then, we can assume that the statement is true for some arbitrary value of n (let's say k) and use this assumption to prove that the statement is also true for n=k+1. This will establish the truth of the statement for all positive integers n.

2. What is the significance of the inequality 1-n^2>0 in this statement?

The inequality 1-n^2>0 is significant because it ensures that n^2 is always less than 1, which means that n is always less than 1. This allows us to make the assumption that 3n-2 is an even integer, since it is equal to 2n+1 where n is a positive integer.

3. Can you provide an example to illustrate this statement?

For example, let n=2. Then, 1-n^2>0 becomes -3>0, which is true. And 3n-2=4, which is an even integer. So the statement is true for n=2.

4. Why is it important to prove this statement?

Proving this statement can help us understand the properties of even integers and their relationship to other mathematical concepts, such as inequalities. It also allows us to make more complex mathematical deductions based on this statement.

5. Are there any practical applications of this statement?

This statement can have practical applications in fields such as computer science, where even integers are commonly used in algorithms and coding. Proving this statement can help ensure the correctness and efficiency of these algorithms.

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