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

76 DETERMINATIONS
Having thus written down ‘multiply’ and ‘square’ in a
sequence (write 1 thereafter). Then starting from 1
backwards, perform the operations of multiplication and
squaring, and finally multiply the resulting quantity by the
first term of the series (i.e., the given sum).¹
Given the geometric series to n terms
a, ar, ar², ..., arⁿ⁻¹,
the object of this rule is to obtain the value of the (n+1)ᵗʰ term,
i. e., arⁿ .
Ex. 108. Some businessman, taking 3 rûpas with him,
went out to make profit. If his capital becomes double after
every month, what will it become after 3 years ?
Here the given sum = 3 rûpas, increase-ratio = 2, and the period = 3
years, i. e., 36 months.
Therefore we have to consider the geometric series, whose first
term = 3, common ratio = 2, and the number of terms = 36.
Proceeding according to the rule, we have
No. of terms of the series | Operation | Write
36 (even) | 36/2 (=18) | square
18 (even) | 18/2 (= 9) | square
9 (odd ) | 9-1 (= 8) | multiply
8 (even) | 8/2 (= 4) | square
4 (even) | 4/2 (= 2) | square
2 (even) | 2/2 (= 1) | square
1 (odd ) | 1-1 (= 0) | multiply
Denoting ‘square’ by s and ‘multiply’ by m, and writing them in a
sequence and 1 in the end, we get
s s m s s s m 1

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¹ Cf. GSS, ii. 94, vi. 311½; MSi, xv. 52-53 (i); SiŚe, xiii. 25; L (ASS),
p. 119, vs. 131; GK, I, p. 127, lines 8-11.