Geometric Progression in Līlāvatī - Revision history

Ramamurthy: Headings modified

2023-08-30T15:08:41Z

Headings modified

← Older revision		Revision as of 20:38, 30 August 2023
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	~~==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''.<ref>{{Cite web\|title=Geometric Progression\|url=https://en.m.wikipedia.org/wiki/Geometric_progression}}</ref> 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.		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''.<ref>{{Cite web\|title=Geometric Progression\|url=https://en.m.wikipedia.org/wiki/Geometric_progression}}</ref> 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==		==Verse 136==
	विषमे गच्छे व्येके गुणकः स्थाप्यः समेऽर्धिते वर्गः ।		विषमे गच्छे व्येके गुणकः स्थाप्यः समेऽर्धिते वर्गः ।

Ramamurthy: Category updated

2023-08-16T12:52:29Z

Category updated

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	<references />		<references />
	[[Category:Mathematics in Līlāvatī]]		[[Category:Mathematics in Līlāvatī]]
			[[Category:General]]

Ramamurthy: Category updated

2023-06-22T10:39:07Z

Category updated

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	==References==		==References==
	<references />		<references />
	[[Category:Mathematics]]		[[Category:Mathematics in Līlāvatī]]
	~~[[Category:~~Līlāvatī]]

Mani: updated Redirecting link to the Hindi Page

2023-06-06T05:46:39Z

updated Redirecting link to the Hindi Page

← Older revision		Revision as of 11:16, 6 June 2023
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	<math>S=\frac{a(r^n -1)}{r-1} = \frac{2(3^7 - 1)}{3-1}= \frac{2(2187-1)}{3-1} = 2186</math>		<math>S=\frac{a(r^n -1)}{r-1} = \frac{2(3^7 - 1)}{3-1}= \frac{2(2187-1)}{3-1} = 2186</math>
	==See Also==		==See Also==
	[https://alpha.indicwiki.in/index.php?title=%E0%A4%B2%E0%A5%80%E0%A4%B2%E0%A4%BE%E0%A4%B5%E0%A4%A4%E0%A5%80_%E0%A4%AE%E0%A5%87%E0%A4%82_%27%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A5%8B%E0%A4%A4%E0%A5%8D%E0%A4%A4%E0%A4%B0_%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A5%87%E0%A4%A2%E0%A4%BC%E0%A5%80%27 लीलावती में 'गुणोत्तर श्रेढ़ी']		[[लीलावती में 'गुणोत्तर श्रेढ़ी']]
	==References==		==References==
	<references />		<references />
	[[Category:Mathematics]]		[[Category:Mathematics]]
	[[Category:Līlāvatī]]		[[Category:Līlāvatī]]

Ramamurthy: column deleted

2023-06-05T22:16:56Z

column deleted

← Older revision		Revision as of 03:46, 6 June 2023
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	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
	\|
	\|		\|
	\|		\|
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	\|M		\|M
	\|2147483648		\|2147483648
	\|
	\|31		\|31
	\|-		\|-
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	\|S		\|S
	\|1073741824		\|1073741824
	\|
	\|30		\|30
	\|-		\|-
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	\|M		\|M
	\|32768		\|32768
	\|
	\|15		\|15
	\|-		\|-
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	\|S		\|S
	\|16384		\|16384
	\|
	\|14		\|14
	\|-		\|-
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	\|M		\|M
	\|128		\|128
	\|
	\|7		\|7
	\|-		\|-
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	\|S		\|S
	\|64		\|64
	\|
	\|6		\|6
	\|-		\|-
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	\|M		\|M
	\|8		\|8
	\|
	\|3		\|3
	\|-		\|-
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	\|S		\|S
	\|4		\|4
	\|
	\|2		\|2
	\|-		\|-
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	\|M		\|M
	\|2		\|2
	\|
	\|1		\|1
	\|}<math>S = a+ar+ar^2+.. +ar^{n-1}</math>		\|}<math>S = a+ar+ar^2+.. +ar^{n-1}</math>

Ramamurthy: /* Introduction */

2023-06-05T22:09:56Z

Introduction

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	\|30		\|30
	\|M		\|M
	\|~~[tel:~~2147483648 ~~2147483648]~~		\|2147483648
	\|		\|
	\|31		\|31
	\|-
	~~\|15~~
	\|S
	~~\|[tel:1073741824 1073741824]~~
	\|
	~~\|30~~
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	\|15		\|15

Ramamurthy: /* Introduction */

2023-06-05T22:07:12Z

Introduction

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	\|S		\|S
	\|[tel:1073741824 1073741824]		\|[tel:1073741824 1073741824]
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			\|15
			\|S
			\|1073741824
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Ramamurthy at 22:00, 5 June 2023

2023-06-05T22:00:04Z

← Older revision		Revision as of 03:30, 6 June 2023
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	==Introduction==		==Introduction==
	In mathematics, a '''geometric progression''', also known as a '''geometric sequence''', is a ~~sequence of~~ 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''.<ref>{{Cite web\|title=Geometric Progression\|url=https://en.m.wikipedia.org/wiki/Geometric_progression}}</ref> 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.		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''.<ref>{{Cite web\|title=Geometric Progression\|url=https://en.m.wikipedia.org/wiki/Geometric_progression}}</ref> 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==		==Verse 136==
	विषमे गच्छे व्येके गुणकः स्थाप्यः समेऽर्धिते वर्गः ।		विषमे गच्छे व्येके गुणकः स्थाप्यः समेऽर्धिते वर्गः ।

Ramamurthy: New Page created

2023-06-05T21:57:27Z

New Page created

New page

==Introduction==
In mathematics, a '''geometric progression''', also known as a '''geometric sequence''', is a sequence of 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''.<ref>{{Cite web|title=Geometric Progression|url=https://en.m.wikipedia.org/wiki/Geometric_progression}}</ref> 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, <math>\frac{n}{2}</math> is called a ‘square’ (S) [Bhaskaracarya’s terminology]. Now beginning with n, continue this process (n-1) for odd and <math>\frac{n}{2}</math> 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.<ref>{{Cite book|title=Līlāvatī Of Bhāskarācārya - A Treatise of Mathematics of Vedic Tradition|publisher=Motilal Banarsidass Publishers|year=2001|isbn=81-208-1420-7|location=New Delhi|pages=110-111}}</ref>

'''Comment''' : Suppose a = 1, n = 31, r =2 . Using the above method we get the table:
{| class="wikitable"
|+
|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 , <math>\frac{30}{2}</math> = 15 , 15-1 = 14 , <math>\frac{14}{2}</math> = 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.
{| class="wikitable"
|+
|
|
|
|
|Index
|-
|30
|M
|[tel:2147483648 2147483648]
|
|31
|-
|15
|S
|[tel:1073741824 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
|}<math>S = a+ar+ar^2+.. +ar^{n-1}</math>

<math>rS = ar+ar^2+..+ar^{n-1}+ar^n</math>

<math>S(r-1)=ar^n-a</math>

<math>S=\frac{a(r^n -1)}{r-1}</math>
==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''':

<math>S=\frac{a(r^n -1)}{r-1} = \frac{2(3^7 - 1)}{3-1}= \frac{2(2187-1)}{3-1} = 2186</math>
==See Also==
[https://alpha.indicwiki.in/index.php?title=%E0%A4%B2%E0%A5%80%E0%A4%B2%E0%A4%BE%E0%A4%B5%E0%A4%A4%E0%A5%80_%E0%A4%AE%E0%A5%87%E0%A4%82_%27%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A5%8B%E0%A4%A4%E0%A5%8D%E0%A4%A4%E0%A4%B0_%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A5%87%E0%A4%A2%E0%A4%BC%E0%A5%80%27 लीलावती में 'गुणोत्तर श्रेढ़ी']
==References==
<references />
[[Category:Mathematics]]
[[Category:Līlāvatī]]