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Proof Example

I am trying to figure out the steps here, not even sure what to really google as the topic to understand it.

My thoughts are

  1. Multiply it out to get $\dfrac{n^2}{2} + \dfrac{n}{2} + (n + 1)$
  2. You then times by $2$ to get ${n^2} + {3n} + {2}$

But then I am stuck. Can someone please walk through the steps that were done to reduce this.

Example taken from how to read and do proofs by Daniel Slow. I know this may be a stupid question just trying to really understand it.

clathratus
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perkss
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    You are just asking about the algebra? But $\frac n2+1=\frac {n+2}2$ is, I think, all you need. – lulu Apr 25 '19 at 19:47

2 Answers2

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First , the common term $\ n+1\ $ is factored out, then two fractions are added by bringing them to the same denominator.

Peter
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  • Could you add a bit more clarity around the n+1 factored out? – perkss Apr 25 '19 at 20:04
  • $\frac{n(n+1)}{2}=(n+1)\cdot \frac{n}{2}$ , so $n+1$ is contained in the first term of the original expression. Formally, the second term can be written as $(n+1)\cdot 1$. Now, use the distributivity-law – Peter Apr 25 '19 at 20:07
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$$ \begin{split} \dfrac{n(n+1)}{2} + (n+1)&= \dfrac{n(n+1)}{2} + \dfrac{2(n+1)}{2} \\ &= \dfrac{n(n+1) + 2(n+1)}{2} \\ &= \dfrac{(n+1)(n+2)}{2} \end{split} $$

mathematics2x2life
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  • So the last two steps are the same information just in a different form. I think is where I got stuck. But I am still not clear how the intermediate step of $(n + 1)(\dfrac{n}{2} + 1)$ occurs – perkss Apr 25 '19 at 20:03
  • I think it's clearer in this form. You are factoring out a $n+1$ that is in common in the two terms in the numerator in the second equality. They do the same thing in your equation. Notice the first term has a $n+1$ in the numerator and the second term is a $n+1$. Factoring this out gives a $n/2$ 'left' in the first term and a $1$ in the second, so factoring out $n+1$ leaves $n/2+1$. Therefore, the term you have in the first line is equivalent to $(n+1)(n/2+1)$. Then $n/2+1= (n+2)/2$, which gives the final equality. It's odd but algebraically correct. – mathematics2x2life Apr 25 '19 at 20:07
  • Thanks so much. Is there a rule for this part or just something you know? /2+1=(+2)/2 – perkss Apr 25 '19 at 20:09
  • This is just simple algebra, get a common denominator: $n/2+1= n/2 + 2/2= (n+2)/2$. – mathematics2x2life Apr 25 '19 at 20:10
  • O yes exactly thank you so much! – perkss Apr 25 '19 at 20:10