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

SERIES IN ARITHMETIC PROGRESSION 69
where, according to vv. 80(ii) to 84,
base = a − d/2,
face = a + (n − ½)d,
and altitude = n.
It may be easily seen that (1) and (2) are the same.
Ex. 103 (i). What is the sum of 5 terms of the series
whose first term is 2 and common difference 3 ? And what
of one-half of a term ?
Ex. 103 (ii). (Also) say (the sum) of one-fifth of a term
of the series whose common difference is 5 and first term 2.
Ex. 104-105. In a leathern oil-bottle (kutapa) full of
oil there occurs a minute hole, and the oil leaks through it.
The bottle has to be carried to a distance of 3 yojanas. If the
wages for the first yojana be 10 paṇas and those for the subse-
quent yojanas successively less by 2 paṇas, what are the wages
for a krośa ?
Sub-rule for finding the first term of a series in A.P. (inter-
preted geometrically by a series-figure), when the common
difference, number of terms, and the sum of the series are
known :
86(i). The sum of the series (gaṇita) as divided by the
number of terms of the series (pada), being diminished by
half the common difference (caya) as multiplied by the
number of terms (gaccha) minus 1, gives the first term of the
series (ādi).¹
That is,
a = s/n − (d/2)(n − 1),
where a, d, n, and s respectively denote the first term, common
difference, number of terms, and the sum of the series.

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¹ Cf. GSS, ii. 74(i) ; vi. 292(i) ; MSi, xv. 48 ; SiŚe, xiii. 23(i); L(ASS),
p. 116, vs. 124 ; GK, I, p. 106, vs. 2. Also see GSS, ii. 73(ii), 76