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

GEOMETRIC SERIES 75
increased by the product of the ‘undestroyed quantity’ and
the common difference and then added to the ‘undestroyed
quantity’, gives the number of terms of the series.
That is, if n denote the integral part of
√[2ds + (a - d/2)²] - (a - d/2)
───────────────────────────────── ,
d
then the number of terms of the series is equal to
s - {(n - 1)d/2 + a}n
n + ─────────────────────────── .
nd + a
The rationale of this rule is as follows :
Let N be the (unknown) number of terms of the series. Then (vide
Rule 85)
s = ( (N - 1)/2 · d + a ) N
or dN² + 2 (a - d/2) N - 2s = 0
giving
N = { √[2ds + (a - d/2)²] - (a - d/2) } / d .
Now let N = n + p/q. Then
n = integral part of { √[2ds + (a - d/2)²] - (a - d/2) } / d ,
and p/q is obtained by the formula (vide Rule 89)
s = (dn + a + a - d) n/2 + (dn + a) p/q
i.e., p/q = [s - {(n - 1)d/2 + a}n] / (nd + a) .
(iii) SERIES IN GEOMETRIC PROGRESSION
Rule for obtaining the amount which a given sum increasing
in a geometric progression would become after a given
period :
94. When the number of terms of the series (i.e., the
number denoting the period) is odd, subtract 1 from it
and write ‘multiply (by the common ratio)’; and when the
number of terms of the series is even, halve it and write
‘square’. (Apply the same rule to the resulting number, and
continue the process till the number reduces to zero.)