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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Figure 1: $latex z=x^2+y^2, n=318$
Figure 2: $latex z=x^4-x^2-y^2+y^4, n=318$
Figure 3: $latex z=x^4-x^2-y^2+y^4, n=315$
Figure 4: $latex z= x^6-x^4-y^4+y^6, n=309$
Figure 5: $latex z=x^6-x^2-y^2+y^6, n=318$
Figure 6: $latex z=x^4-x^2+y^2-y^4, n=310$
