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

82 DETERMINATIONS
Ex. 113. In a gamble two (persons) alternately won
30 , 10 , 100 and 8 casts of dice (with stake-moneys) be-
ginning with 9 and increasing (successively) by 6. Say who
is the winner.
Ex. 114. If the casts of dice (alternately won by the two
persons) be 7, 3, 9 and 12, and the first term and com-
mon difference (of the series formed by the stake-moneys)
as stated before, then say after calculation who wins, if you
know (the method).
Ex. 115. In a gamble two (persons) alternately won
from each other 4, 3, 2 and 2 casts of dice (with stake-moneys)
beginning with 1 and increasing (successively) by 6. Say
who is the winner.
(v) SERIES OF SQUARES, CUBES, AND SUCCESSIVE SUMS OF
NATURAL NUMBERS
r = n r = n
Rule for finding the sum of ∑ r + n² + n³ and of ∑ r² :
r = 1 r = 1
102. The number of terms plus one, as multiplied by
twice the number of terms plus one, being (further) multiplied
by half the number of terms, gives the sum of (i) the sum of a
series of natural numbers (from 1 up to the given number of
terms), (ii) the square of the number of terms, and (iii) the
cube of the number of terms.
That divided by 3 gives the sum of a series of squares of
natural numbers.
That is
r = n
(1)¹ ∑ r + n² + n³ = [(2n + 1) (n + 1) n] / 2 ,
r = 1
r = n
(2)² ∑ r² = [(2n + 1) (n + 1) n] / (2 × 3) .
r = 1

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¹ For an alternative formula, see GSS, vi. 296; GK, I, p. 116, lines 3-4.
² Cf. Ā, ii. 22; BrSpSi, xii. 20 (i); SiŚe, xiii. 22 (i); L (ASS), I, p. 113, vs. 19 (i); GK, I, p. 117, lines 1-2.