Two exceedingly simple sums related to triangular numbers


This note records some elementary arithmetic pertaining to triangular numbers for bālabodhana. In our youth we found that having a flexible attitude was good thing while obtaining closed forms for simple sums: for some sums geometry (using methods of proofs pioneered by Āryabhaṭa which continued down to Nīlakaṇṭha Somayājin) was the best way to go; for others algebra was better. The intuition was in choosing the right approach for a given sum. We illustrate that with two such sums.

Sum 1 Obtain a closed form for the sum: $latex displaystyle sum_{j=1}^{n} (2j-1)^3$

These sums define a sequence: 1, 28, 153, 496, 1225…
Given that we can mostly only visually operate in 3 spatial dimensions, our intuition suggested that a cubic sum as this is best tackled with brute-force algebra with the formulae for individual terms derived by Āryabhaṭa and his commentators. Thus we have:

$latex displaystyle sum_{j=1}^{n} (2j-1)^3 = sum_{j=1}^{n}…

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It is absolutely clear that this world and its contents are purely mental and nothing solid in it. To come out of suffering entirely you should know your mind thoroughly.

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