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I thought about some random number-stuff recently and I just summed some odd number and realised that

$\displaystyle\sum_{k=1}^{n}2k-1=n^2$ for $n\in\mathbb{N}^+$

I was wondering how you'd prove this and why this holds true.

0xff
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    Welcome to Mathematics SE. Take a tour: https://math.stackexchange.com/tour. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context by stating what you understand about the problem, what you've tried so far, etc.; both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance: https://math.meta.stackexchange.com/q/9959 . – Kavi Rama Murthy Oct 05 '21 at 12:27
  • An internet search for "sum of odd numbers" ought to give several results. – Arthur Oct 05 '21 at 12:27
  • One method to prove it is induction. You can also use Gauss's pairing approach he used to calculate the sum of the numbers from $1$ to $100$ as a very young school-boy. – Peter Oct 05 '21 at 12:30
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    To visualize, Draw a $1=1\times 1$ square, then add $3$ to get a $1+3=2\cdot 2$ square, then add $5$ to get a $1+3+5=3\cdot 3$ square... as seen here.

    $$\begin{aligned}{} \color{blue}{\Rule{10mm}{10mm}{1mm}} \color{blue}{\Rule{10mm}{10mm}{1mm}} \color{blue}{\Rule{10mm}{10mm}{1mm}}\[-1em] \color{red}{\Rule{10mm}{10mm}{1mm}} \color{red}{\Rule{10mm}{10mm}{1mm}} \color{blue}{\Rule{10mm}{10mm}{1mm}}\[-1em] \color{black}{\Rule{10mm}{10mm}{0mm}} \color{red}{\Rule{10mm}{10mm}{0mm}} \color{blue}{\Rule{10mm}{10mm}{0mm}} \end{aligned} $$

     – Vepir Oct 05 '21 at 12:39
  • @Vepir Why are you answering in the comment section of a closed question? – Arthur Oct 06 '21 at 11:24
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    Why would they not? – 0xff Oct 06 '21 at 14:35
  • @Arthur It is more of a pointer (and a summary) to an already existing linked answer of the linked duplicate. Though, since the answer is so simple, it is indeed almost indistinguishable from the answer. – Vepir Oct 06 '21 at 15:39
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    @Jadefalke Comments are not meant for answering questions. – Vepir Oct 06 '21 at 15:40
  • @Jadefalke That. And also, answers to closed questions are unwanted on this site, regardless of where on the post the answers are placed. – Arthur Oct 06 '21 at 16:58

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