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

68 DETERMINATIONS
Rule for finding the face of the series-figure corresponding to
the given series :
84. Having constructed the series-figure (for altitude
unity) in this manner, one should determine the face for the
desired altitude (i.e., for the desired number of terms of the
series) (by the following rule) :
The face (for altitude unity) minus the base (for altitude
unity), multiplied by the desired altitude, and then increased
by the base (for altitude unity), gives the face (for the desired
altitude).
This rule, on simplification, reduces to the following formula :
face for altitude n = a + (n - ½) d.
Rule for finding (i) the sum of a series in A. P. (interpreted
geometrically by a series-figure), and (ii) the area of the
corresponding series-figure :
85. The common difference as multiplied by one-half
of the number of terms minus one, being increased by the first
term, and then multiplied by the number of terms, gives the
sum of the series.¹
And the area of the (corresponding) series-figure is equal
to the product of one-half of the sum of the base and the face,
and the altitude.
That is, the sum of the series
a + (a + d) + (a + 2d) + ... to n terms
is equal to { (n - 1)/2 d + a } n ; (1)
and the area of the corresponding series-figure is equal to
(base + face) / 2 × altitude, (2)

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¹ Cf. Ā, ii. 19 ; GSS, ii. 61, vi. 29.) ; GK, I, 105, lines 12-13. The
rule of the GK is literally the same as above.
Brahmagupta (BrSpSi, xii. 17) states the result in the modern form :
Sₙ ≡ a + (a + d) + (a + 2d) + .. to n terms = (n/2)[2a + (n - 1)d].
The same form is given in GSS, ii. 62; MSi, xv, 47 ; Siśe, xiii. 20 ; L (ASS)
p. 114, vs. 121 ; GK, I, p. 105, vs. 1.