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In the following sutra another rule is given to find out the sum of a series in G. P.1
समदलविषमखरूपो गुणगुणितो वर्गताडितो गच्छः । रूपोनः प्रभवघ्नो व्येकोत्तरभाजितः सारम् ॥
"The number of terms in the series is caused to be marked (in a separate column) by zero and by one (respectively) corresponding to the even (value) which is halved and to the uneven (value from which one is substracted till by continuing these processes zero is ultimately reached), then this (representative series made up of zero and one is used in order from the last one thcrc in, so that this one multiplicd by the common ratio is again) multiplied by the common ratio (wherever one happens to be the denoting item) and multiplied so as to obtain the square (wherever zero happen in the invins iv...). This the result of this (operation) is diminished by one and (is then) multiplicd by the first term, and (is then) divided by the common ratio lessened by one it becomes the sum of the series)." Example :
स्वर्ण द्वयं गृहीत्वा त्रिगुणधनं प्रतिपुर समार्जयति ।
यः पुरुषोऽष्टन गर्यां तस्य कियद्वित्तमाचक्ष्व ।। *Having obtained 2 gold coins (in some city), a man goes on from city to city, earning (everywhere) three times (of what he earned immediately before). Say how much he will make in the eighth city." Solution : Here n = 7, r = 3, a = 2
7 an odd number, hence one is subtracted from it and also it is denoted by one. 2-1 = 6 = an even number, hence it is divided by 2 and 0 denotes it
ہے
و
= 3 = an odd number, it is diminished by one and I denotes it
ہ برا بردم
-- 1 = 2 = an even number, it is divided by 2 and 0 denotes it
= 1 - an odd number, it is diminished by one and 1 denotes it 1-1=0 =, where the operation ends.
Now the whole is put in the side column. Since in the column, 1 is in the last hence it is 11 multiplied by the common ratio 3, then comes zero so 3 is squared and we get 39, then comes 1 above lo it so it is multiplied by » i.e. we get 33, then comes zero above it so it is squared and we get 39. then in the end there is one above it so it is multiplied by 3 and get 3?. So the guradhana
arn 2 37 = 2 X 2187 = 4374 coins, will be the amount obtained by the man in eighth city.
The rules for finding out the last term and the sum of series in G. P. have also been given in nza3 There are other sutras in which rules have been given to find out the first term, common ratio and the number of terms of the series in G. P. 4, 5, 6, 7.
94
90
1. GSS 2. GSS 3. GSS 4. GSS
GSS 6. GSS 7. GSS
2 2 2 2
98
101
103
जन प्राच्य विद्याएं
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