An elementary extension attempt

Who: J2
When: just after brushing his teeth for bed
Where: bedroom

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A while ago, J2 realized that adding consecutive odd whole numbers would give him squares. For example, $1+3+5+7+9 = 25.$

He loves to ask variations on this sum of odds question. Tonight, he came up with a new idea:

what is the sum of odds up to 10?

We discussed and he clarified that he wanted 10 included somehow, not just the equation above. How would you extend the idea of adding consecutive odd numbers?

Our idea
The odd sum and squares relationship looks like this:

$$ 1 + 3+ \cdots + (2n - 1) = n^2$$

This isn't how J2 thinks of it yet. He focuses on the largest term $m$ and then, through sequential calculations, forms $m+1$, then $(m+1)/2$ and then $\left(\frac{m+1}{2}\right)^2$.

His idea tonight for answering his own question was to apply the same procedure (formula) to 10. In essence, what he was doing was saying:

• I know what it means to "add odd numbers up to $m$" when $m$ is odd.
• I know that the result is $\left(\frac{m+1}{2}\right)^2$ in that case.
• I can then use that as a definition of what it means to "add odd numbers up to $m$" when $m$ is not odd!

I helped with the final squaring calculation, giving the answer: 30.25 is the sum of odd integers up to 10.

Credits
My ability to now include $\TeX$ equations is thanks to Sachin Shanbhag and Steve Holden.