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

74 DETERMINATIONS
Sub-rule for finding the common difference of a series in A.P.
(interpreted symbolically), when the first term, the number
of terms which is partly integral and partly fractional, and
the sum of the series are known :
91. The sum of the series (dhana) as diminished by the
product of the first term and the number of terms, should be
divided by the sum of the series in which the first term and
common difference are each unity and the number of terms
is equal to the given number of terms minus 1. The result
which is thus obtained is the common difference.
That is, if n+p/q be the number of terms, then
s - a(n+p/q)
d = ------------ ,
S
where S is the sum of (n+p/q - 1) terms of the series 1+2+3+ ..., i. e.,
(n - 1)n p
-------- + ― n.
2 q
(Vide supra, Rule 89).
Sub-rule for finding the number of terms of a series in A.P.
which is partly integral and partly fractional, when the first
term, the common difference, and the sum of the series are
known, the series being interpreted symbolically :
92-93. To the sum of the series (dhana) as multiplied
by twice the common difference (caya), add the square of (the
difference of) the first term minus half the common difference:
of that obtain the nearest (integral) square root. That (square
root), diminished by the square root of the previous square
(i.e., the square of the difference of the first term minus half
the common difference), (then) divided by the common
difference, and (then) rid of its fractional part (vikala), is
the so called ‘undestroyed quantity.’
That (‘undestroyed quantity’) lessened by 1, being
multiplied by half the common difference, (then) increased
by the first term (mukha), and (then) multiplied by the ‘un-
destroyed quantity’, should be subtracted from the sum of the
series (gaṇita). That being divided by the first term as