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I'm trying to express the next two consecutive odd numbers in terms of x. How do go about this?

Precisely, given a number $x$ I want two smallest consecutive odd numbers $m,n$ such that $x<m<n$.

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  • $\begingroup$ Be aware that you wrote two very different things in your question and your body. Are you trying to express the next two consecutive odd numbers in terms of x, or are you trying to express x in terms of the next two consecutive odd numbers? If $x$ is, say $5$, do you consider "the next two consecutive odd numbers" to be $5$ and $7$ or do you consider the next two consecutive odd numbers to be $7$ and $9$? $\endgroup$ – JMoravitz Oct 19 '15 at 2:32
  • $\begingroup$ My error. I'm trying t express the next two consecutive odd numbers in terms of x. Edited to reflect. $\endgroup$ – user214824 Oct 19 '15 at 2:45
  • $\begingroup$ That still doesn't answer the question of what exactly you mean by "next two consecutive odd numbers" is. If $5$ is your number, do you want $7$ and $9$? or do you want $5$ and $7$ (despite $5$ not being bigger than $5$). $\endgroup$ – JMoravitz Oct 19 '15 at 2:47
  • $\begingroup$ @JMoravitz Yes, that is it. 7 and 9. $\endgroup$ – user214824 Oct 19 '15 at 2:48
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Presumably $x$ is an integer. We have two cases: either $x$ is even or $x$ is odd.

In the case that $x$ is odd, then $x+2$ and $x+4$ will both be odd and will be the next two consecutive odd numbers.

In the case that $x$ is even, then $x+1$ and $x+3$ will both be odd and will be the next two consecutive odd numbers.

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Let $x$ an integer, then the next two odd integers are $$2\times\Bigl\lfloor \frac{x+1}{2}\Bigr\rfloor + 1 \qquad\text{and}\qquad 2\times\Bigl\lfloor \frac{x+1}{2} \Bigr\rfloor + 3.$$

Indeed, if $x = 2n$ where $n\in\mathbb{Z}$, then $$ 2\lfloor (x+1)/2\rfloor + 1 = 2 \lfloor n + 1/2 \rfloor + 1 = 2n + 1 $$ and if $x = 2n + 1$, then $$ 2\lfloor (x+1)/2 \rfloor + 1 = 2\lfloor n + 1 \rfloor + 1 = 2n + 3. $$

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  • $\begingroup$ I can't understand this due to my level of comprehension. But someone else definitely will. $\endgroup$ – user214824 Oct 19 '15 at 2:53
  • $\begingroup$ Maybe doesn't you know the symbol $\lfloor \cdot \rfloor$. If $x$ is a real number, $\lfloor x \rfloor$ is the biggest integer less or equal to $x$. For example $\lfloor 2.7 \rfloor = 2$, $\lfloor -3.8 \rfloor = -4$ and $\lfloor x \rfloor = x$ for $x$ integer. $\endgroup$ – Éric Guirbal Oct 19 '15 at 3:01

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