Geometric Progression in Līlāvatī
Introduction
In mathematics, a geometric progression, also known as a geometric sequence, is a sequenceof non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.[1] For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.
Verse 136
विषमे गच्छे व्येके गुणकः स्थाप्यः समेऽर्धिते वर्गः ।
गच्छक्षयान्तमन्त्याद् व्यस्तं गुणवर्गजं फलं यत्तत् ।
व्येकं व्येकगुणोद्धृतमादिगुणं स्यात् गुणोत्तरे गणितम् ॥
If n, the number of terms in a Geometric Progression (G.P) is odd, then (n-1) is called a ‘multiplier’ (M) and if it is even, is called a ‘square’ (S) [Bhaskaracarya’s terminology]. Now beginning with n, continue this process (n-1) for odd and for even until 0 is reached.Then keep the common ratio (C.R) r against 0 and start writing M or S in the opposite direction. Carry out the operations and then subtract 1 from the final result and divide it by (r-1). The result is the required sum.[2]
Comment : Suppose a = 1, n = 31, r =2 . Using the above method we get the table:
31 | 30 | 15 | 14 | 7 | 6 | 3 | 2 | 1 | 0 |
M | S | M | S | M | S | M | S | M |
Beginning with 31 , we write 31-1 = 30 , = 15 , 15-1 = 14 , = 7 …. until we reach 0. Then in the second line we begin with M and alternately write M and S.
Now we perform the indicated operations. M = multiplier, S = square as shown below.
Index | ||||
30 | M | 2147483648 | 31 | |
15 | S | 1073741824 | 30 | |
15 | S | 1073741824 | 30 | |
14 | M | 32768 | 15 | |
7 | S | 16384 | 14 | |
6 | M | 128 | 7 | |
3 | S | 64 | 6 | |
2 | M | 8 | 3 | |
1 | S | 4 | 2 | |
0 | M | 2 | 1 |
Example
आदिर्द्वयं सखे वृद्धिः प्रत्यहं त्रिगुणोत्तरा ।
गच्छ सप्तदिनं यत्र गणितं तत्र किं वद ॥
If a = 2, r = 3, n = 7, O friend , what is the sum ?
Comment:
See Also
लीलावती में 'गुणोत्तर श्रेढ़ी'
References
- ↑ "Geometric Progression".
- ↑ Līlāvatī Of Bhāskarācārya - A Treatise of Mathematics of Vedic Tradition. New Delhi: Motilal Banarsidass Publishers. 2001. pp. 110–111. ISBN 81-208-1420-7.