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

LOGISTICS 5
LOGISTICS (Parikarma)
(1) Addition (saṅkalita)
Rule for finding the sum of a series of natural numbers :
14(i). When the first term (ādi) and the common differ-
ence (caya) of a series (in arithmetic progression) are (each)
unity, the sum (saṅkalita) is equal to half the number of terms
(pada) multiplied by the number of terms plus one.¹
That is,
Sₙ = 1 + 2 + 3 + ... + n = n(n + 1) / 2 .
Rule for finding the number of terms of a series of natural
numbers when the sum is given :
14(ii). The number of terms (gaccha) is equal to the
(integral) square root of twice the sum of the series, which
must be the same as the residue left (after the extraction
of the square root).
That is,
n = integral part of √(2 Sₙ) .
Ex. 1. What are the sums of 1 to 10, each multiplied
by 10, terms of the series whose first term and common
difference are each unity ? Also, from those sums quickly say
the number of terms of the (various) series.
*‘1 to 10, each multiplied by 10’ means 10, 20, 30, 40, 50, 60, 70, 80,
90 and 100.
Alternative rules :
15. Or, the sum of a series (of natural numbers) is
equal to one-half of—the square of the number of terms plus
the number of terms.
And that (sum) multiplied by 8, (then) increased by 1,
(then) reduced to its square root, (then) diminished by 1,

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¹ Cf. GK, I, p. 114, lines 15-16.