Contribution of Mahaviracharya in the

Author(s): R S Lal
Publisher: Z_Deshbhushanji_Maharaj_Abhinandan_Granth_012045.pdf

Page 5

Partial Sums
The sum of any part of a series is known as the partial sum of the series. In the following verse, the method is given for finding the partial sum of a given series :1
"(Take) the chosen off number of terms as combined with the total number of terms (in the series) and (take) also your own chosen off number of terms (simply) diminish (each of) these (resulting products) and these (resulting quantities) when multiplied by the remaining number of terms (respectively), give rise to the sum of the remainder series and to the sum of the chosen off part of the series (in order)."
Symbolically. Vyutkalita-S
and the sum of the chosen part=S,
सपदेष्टं स्वष्टमपि व्यकं दलितं यातं समुखम् । शेषेष्ट गच्छ गुणितं व्युत्कलितं स्वेष्टवित्तं च ॥
[Pd+a] p.
where p is the number of terms of the chosen part of the series. Another form of the same formula is given in a different verse.2
Example:
In the following sutra is given the rule for finding the sum of a series in arithmetic progression in which the common difference is either positive or negative :
1.
GSS
2. GSS
3. GSS
4. GSS
जन प्राच्य विद्याएं
[마을 2=1
"The first term is either decreased or increased by the product of the negative or the positive common difference and the quantity obtained by halving the number of terms in the series as diminished by one. (Then), this is (further) multiplied by the number of terms of the series and (thus), the sum of series of terms in arithmetical progression with positive or negative common difference is obtained."
1
Symbolically, S("="+a 11d+a) n
where a, d, n and S have their usual meanings.
2
2
6
6
-d+a] (n-p)
व्ये काप दोनाधिकचयघातोनान्वितः पुनः प्रभवः । गच्छाभ्यस्तो हीनाधिकचयसमुदाय संकलितम् ॥
32
33
165
165
चतुरुत्तरदश चादिर्हीनच यस्त्रीणि पञ्च गच्छः किम् । द्वावादिवृद्धिचयः षट् पदमष्टी घनं भवेदत्र ||
106
107
290
291
६७