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

84 DETERMINTAIONS
That is
m = n r = m r = n r = n n(n+1)² (n+2)
∑ ∑ r + ∑ r² + ∑ r³ = —————————— .
m = 1 r = 1 r = 1 r = 1 4
Ex. 118. Friend, if you know then say after calculation
the sum of (i) the sum of successive sums of the first 6 natural
numbers, (ii) the sum of the squares of the first 6 natural
numbers, and (iii) the sum of the cubes of the first 6 natural
numbers.
(vi) SERIES OF SQUARES AND CUBES, ETC. OF THE TERMS
OF AN ARITHMETIC SERIES
Rule for finding the sum of the squares of the terms of the
r = n
series ∑ {a+(r-1)d} :
r = 1
105. The sum of the ( arithmetic ) series with
twice the common difference, when multiplied by the first
term and (then) increased by the sum of the squares of the
natural numbers ranging from 1 to one-less the number of
terms, as multiplied by the square of the common difference,
gives the sum of the squares of the terms of the (given arith-
metic) series.¹
That is,
r = n
∑ {a+(r-1)d}²
r = 1
= [a+(a+2d) +(a+4d) +...to n terms] × a

• [1² + 2² + 3² +...+ (n-1)²] × d².
  Ex. 119. Tell me the sum of the squares of (the first)
  six terms of the (arithmetic) series whose first term is two and
  common difference three.
  Rule for finding the sum of the sums of series of natural num-
  bers each beginning with 1, whose number of terms are the
  terms of a given arithmetic series :

106. Having obtained the sum of the squares of the
     terms of the (arithmetic) series with (given) first term,

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¹ The same rule occurs in GK, I, p. 119, lines 7-8, and p. 120,
lines 1-2. For further rules, see GSS, vi. 298, 299.