Geometric Progression in Līlāvatī

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. 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, ${\frac {n}{2}}$ is called a ‘square’ (S) [Bhaskaracarya’s terminology]. Now beginning with n, continue this process (n-1) for odd and ${\frac {n}{2}}$ 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.

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 , ${\frac {30}{2}}$ = 15 , 15-1 = 14 , ${\frac {14}{2}}$ = 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
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

$S=a+ar+ar^{2}+..+ar^{n-1}$

$rS=ar+ar^{2}+..+ar^{n-1}+ar^{n}$

$S(r-1)=ar^{n}-a$

$S={\frac {a(r^{n}-1)}{r-1}}$

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:

$S={\frac {a(r^{n}-1)}{r-1}}={\frac {2(3^{7}-1)}{3-1}}={\frac {2(2187-1)}{3-1}}=2186$

See Also

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