पाटीगणित (श्रीधराचार्य - प्राचीन संस्कृत टीका एवं प्रो. कृपाशंकर शुक्ल सविस्तार व्याख्या)

6 LOGISTICS
and (then) halved, is the number of terms (in the series).
That is,
Sₙ ≡ 1 + 2 + 3+.... + n = (n² + n) / 2 ,
and n = (√(8 Sₙ + 1) - 1) / 2 .
(2) Subtraction (vyavakalita)
Rule for finding the remainder accruing on subtracting one
series of natural numbers (‘subtrahend series’) from another
series of natural numbers (‘minuend series’) :
16. Having added (nidhāya) the number of terms of the
subtrahend series (vyavakalita-pada) plus one to the number of
terms of the minuend series (saṅkalita-pada), multiply that (sum)
by the difference of the number of terms (of the two series) :
that (product), when halved, becomes the residue of subtrac-
tion (of the given series).
That is,
Sₙ - Sₘ = (n - m) [n+ (m+1)] / 2 ,
where n > m, and Sₖ stands for
1 + 2 + 3 +...+ k .
Ex. 2. What are the respective remainders obtained
when the sums of 1 to 10, each multiplied by 10, terms of the
series whose first term and common difference are (each)
unity are severally subtracted from the sum of 100 terms (of
the same series) ?
Rule for finding the number of terms of the subtrahend series,
when the difference of the minuend and subtrahend series as
well as the number of terms of the minuend series is given :
17. Having subtracted the residue of subtraction (i.e.,
the difference of the minuend and subtrahend series) from
the sum of the minuend series, and multiplied the remainder
(obtained) by 2, the square root thereof, which must be equal