________________
which can be generalised as S = +
-1 +S Now we shall discuss the contribution of Mahavira in the development of the series which can be put in another category called miscellaneous series. This work no doubt, is quite voluminous and it can be said without any hesitation that no other Hindu mathematician contributed so much.
In the following stanza a rule is given for finding the sum of the squares of natural numbers. He has not given any formula for the sum of natural numbers like others.
सेकेष्टकृतिविघ्ना सैकेष्टोनेष्टदलगुणिता । कृतिघनचितिसंघातस्त्रिकभक्तो वर्गसंकलितम् ।।
Algebraically, if
n = number of terms and
n(n+1)
-
sum of first n natural nos.
En = sum of the squares of n natural nos.
then
} [ 2 (n+1)* — (n + 1) ] = = [ ** + n° + " (n!! ]} =
= nx(n+1) X(2n+1)
In the following sutra a rule is given for finding the sum of the squares of numbers which are in A. P. This is most general form of the rule which can be applied broadly.2
द्विगुण कोनपदोत्तरकृतिहतिषष्ठांशमुखचयहतयुतिः ।
व्येकपदघ्ना मुखकृतिसहिता पदताडितेष्टकृतिचितिका ॥ Algebraically, if a = first term, d = common diff.
n= number of terms and s = sum of the squares of the terms which are in A. P. then
S = E[a + (n—1) a 1 = n [{{20 ) d + ad} (n–1 + qe ] which can easily be substantiated by taking LHS. i.e. {a + (n-1) d]' = [(a-d) + 2 nd (
a d) + na da] = n(a-d)2 + 2 d (a-d) En + d2 En2 we know that sn= n(n+1) and 5 ne = n(n+1) (2n+1)
2
Hence by substituting these values we get the result
1. GSS 2. GSS
6 6
167 167
296 208
a seu faang
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