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

72 DETERMINATIONS
the number of terms minus 1, as diminished by 1 and multi-
plied by the number of terms. Thus is obtained the first term
of the series (âdi).
That is
a = [ {(n² - n) / 2}(a + d) - s ] / [ {(n - 1) / 2 - 1} n ] ,
where the symbols have their usual meanings.
We know (vide supra Rule 85) that
s = { (n - 1)/2 · d + a } n
= [ (n - 1)/2 · (a + d̅ - a) + a ] n
= {(n² - n) / 2}(a + d) - { (n - 1)/2 - 1 } na
∴ a = [ {(n² - n) / 2}(a + d) - s ] / [ { (n - 1)/2 - 1 } n ] .
(ii) SERIES IN ARITHMETIC PROGRESSION
( SYMBOLICAL INTERPRETATION )
According to the symbolical interpretation, the series
a + (a + d) + (a + 2d) + ... to (n + p/q) terms
means the sum of n terms together with the p/qᵗʰ part of the (n + 1)ᵗʰ
term.
Rule for finding the sum of a series in A.P. (interpreted
symbolically), when the number of terms is partly integral
and partly fractional, the first term, common difference,
and the number of terms being known :
89. The common difference (caya) as multiplied by the
integral part of the number of terms (nirvikalapada) should be
increased by the first term (âdi), and the result (obtained)
should be kept undestroyed (at one place). The same result
(written in another place) being increased by the first term
(mukha), (then) diminished by the common difference, (then)
multiplied by one-half of the integral part of the number of