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

SERIES IN ARITHMETIC PROGRESSION 71
number of terms of the series (gaccha).¹
That is,
n = [√(8ds + (2a - d)²) - 2a + d] / 2d ,
where a, d, n and s have their usual meanings.
We know (vide supra Rule 85) that
s = [((n - 1) / 2) d + a] n ,
∴ dn² + (2a - d)n - 2s = 0.
On solving this quadratic for n, we get²
n = [√(8ds + (2a - d)²) - 2a + d] / 2d ,
taking the positive sign of the radical, because n is positive.
Hence the above rule.
Sub-rule for finding the first term of a series in A.P. (inter-
preted geometrically by a series-figure), when the sum
of the series, the number of terms, and the sum of the first
term and common difference are known :
88. Having subtracted the sum of the series (phala)
from the mixed amount (miśra-dhana) (i.e., the sum of the first
term and common difference) as multiplied by one-half of (the
difference of) the number of terms squared minus the number
of terms, divide the residue by one-half of (the difference of)

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¹ Cf. BrSpSi, xii. 18 ; GSS, ii. 70. Also see GSS, ii. 69.
² Âryabhaṭa I (Â, ii. 20) puts the result in the following form :
n = ½ [ {√(8ds + (2a - d)²) - 2a} / d + 1 ].
Âryabhaṭa II (MSı, xv. 50), Bhâskara II (L, p. 118, vs. 128) and
Nârâyaṇa (GK, I, p. 107, lines 4-7) put it in the form :
n = [√(2ds + (a - d/2)²) - a + d/2] / d .
Śrîpati (SiŚe, xiii. 24) puts it in the form :
n = √[ s / (d/2) + ((a - d/2) / d)² ] - (a - d/2) / d .
Indological Truths