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

SERIES OF SQUARES, CUBES AND SUCCESSIVE SUMS 83
Ex. 116 (i). Say what is the sum of (i) the sum of
the first five natural numbers, (ii) the square of 5, and (iii) the
cube of 5.
Ex. 116 (ii). Also (say), if you know, the sum of the
squares of 5 terms of the series whose first term and common
difference are each unity.
Rule for finding the sum of ∑(r = 1)^(r = n) r³ and of ∑(m = 1)^(m = n) ∑(r = 1)^(r = m) r :
103. One-half of what is obtained by adding the
number of terms to the square of the number of terms, when
multiplied by itself, gives the sum of the cubes of natural
numbers (from 1 up to the given number to terms) ; and when
multiplied by the number of terms plus 2 and divided by 3,
gives the sum of the successive sums of those natural numbers.
That is,
(1)¹ ∑(r = 1)^(r = n) r³ = {(n² + n) / 2}² ;
(2)² ∑(m = 1)^(m = n) ∑(r = 1)^(r = m) r = [((n² + n) / 2)(n + 2)] / 3 .
Ex. 117. Friend, quickly say what is the sum of the
cubes of 10 terms of the series whose first term and common
difference are each unity ; and also the sum of the successive
sums of those terms.
Rule for finding the sum of ∑(m = 1)^(m = n) ∑(r = 1)^(r = m) r + ∑(r = 1)^(r = n) r² + ∑(r = 1)^(r = n) r³ :
104. The number of terms, as multiplied by the square
of (the sum of ) the number of terms plus one, when (further)
multiplied by the number of terms plus two and divided by
four, gives the sum of (i) the sum of successive sums of natural
numbers (from 1 up to the given number of terms), (ii) the
sum of squares of those natural numbers, and (iii) the sum of
cubes of those natural numbers.
¹ Cf. Ā, ii. 22(ii) ; BrSpSi, xii. 20(ii) ; GSS, vi. 301 ; SiŚe, xiii. 22(ii);
L(ASS), I, p. 113, vs. 119(ii); GK, I, p. 117, lines 3-4.
² Cf. Ā, ii. 21; BrSpSi, xii. 19; SiŚe, xiii. 21; L(ASS), I, p. 112, vs. 117.
Indological Truths