# Is n^2+n+3 Always Odd for Any Natural Number n?

• mathstudent88
In summary, if n is a natural number, then n^2+n+3 is always odd. This can be proven by observing that when n is even, n^2+n+3 can be factored into 2(2k^2+k+1)+1, where (2k^2+k+1) is a natural number. Similarly, when n is odd, n^2+n+3 can be factored into 2(2j^2+3j+2)+1, where (2j^2+3j+2) is a natural number. Therefore, n^2+n+3 is always odd regardless of whether n is even or odd.
mathstudent88
If n is a natural number then n2+n+3 is odd.

This is what I have and wanted to know if I was doing it right or not:

Let n be a member of the natural numbers. If n is even, then n=2k, k member of natural numbers, and n2+n+3
=(2k)2+2k+3
=4k2+2k+3
= 2(2k2+k+1)+1, where (2k2+k+1) is a member of the natural numers. This means that when n is even, n2+n+3 is odd.

If n is odd, then n=2j+1 where j is a member of the natural numbers
and n2+n+3
=(2j+1)2+(2j+1)+3
=4j2+4j+1+2j+4
=4j2+6j+5
=2(2j2+3j+2)+1, where (2j2+3j+2) is a memeber of the natural numbers. This means that when n is odd, n2+n+3.

Is this ok? Thanks for the help!

It looks ok, but it's overly complicated. A simpler argument is that n^2+n=n(n+1) is always even (because clearly one of n, n+1 is), therefore, n^2 + n + 3 is always odd.

I would say that what you give is a perfectly good proof and because it is yours it is the one you should submit. Of course, you should then be aware of Preno's simpler proof.

## 1. What are natural numbers?

Natural numbers are a set of positive integers starting from 1 and extending to infinity. They are used to count objects or to represent quantities.

## 2. What is a proof in mathematics?

A proof is a logical argument that shows a statement or theorem is true. It is used to demonstrate the validity of a mathematical statement using established axioms, definitions, and previously proven theorems.

## 3. Why is it important to prove the properties of natural numbers?

Proving the properties of natural numbers is important in mathematics as it provides a solid foundation for more complex theories and allows for the development of new theorems and formulas. It also helps to ensure that mathematical statements and calculations are accurate and reliable.

## 4. What are some properties of natural numbers?

Some properties of natural numbers include closure, commutativity, associativity, distributivity, and identity. For example, the closure property states that when adding, subtracting, multiplying, or dividing two natural numbers, the result will always be a natural number.

## 5. How can I prove a property of natural numbers?

To prove a property of natural numbers, you can use mathematical induction, which is a method of proof that involves showing the property holds for a base case (usually 1) and then using the assumption that it holds for one number to show that it also holds for the next number. This process is repeated until the desired property is proven for all natural numbers.

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