On The Peculiar Formula for Finding
The Sum of An A. P. in India

सारांश

प्रस्तुत आलेख में समानान्तर श्रेणी के स् पदों का योगफल निकालने के विशिष्ट सूत्र पर गणितीय विवेचन किया गया है। इस सूत्र को आचार्य यतिवृषभ ने दिया है।

”’The formula for finding the nth term `1` of an arithmetic progression (abbreviated as A.P.) is given by

1 = a + (n-1)d [1]

and the formula for finding its sum `S’ up to n terms is given by

S = [(n-1) d + 2a ] n/2 [2]

where `a’ is the first term and `d’ is the common difference in the A.P.

Consider the following A.P.

5,
13, 21, 29,
37, 45, 53, 61, 69,
77, 85, 93, 101, 109, 117, 181, 189, 197
133, 141, 149, 157, 165, 173, 181, 189, 197
205, 213, 221, 229, 237, 245, 253, 261, 269, 277, 285,
293, 301, 309, 317, 325, 333, 341, 349, 357, 365, 373, 381, 389,

Here a = 5, d = 8, and n = 49. The A.P. is comprised in seven groups, each succeeding group containing two more terms. The number of terms in each group is odd.

The Tilloyapannafi1 (abbreviated as TP, `A suggestion on the three region of the universe’ is an ancient text composed by yatvrsabha in nine chapters in Prakrta. It deals with Jaina cosmography and also with many topics of religious and cultural interest. Not much is known about the life of Yativrsabha for except that he was a Digambara Jaina ascetic.

He belonged to the canonical class of the Jaina school of Indian mathematics. Scholars are not agreeing in opinion about his time. Some date circa 176 A.D. is assigned to him by one scholar2 and some period between 473 A.D. and 609 A.D. to him by another.

एक्कोणमवणि इंदयमद्धिय वग्गेज्ज मूल संजुत्तं।
अट्ठ—गुणं पंच—जुदं, पुढ़िंवदय ताडिदम्मि पुढ़विधणं।।[ TP, v. 2.65, p. 159]

“The terms (indayas (indrakas in Sanskrit, central cavaties) in the verse, m) as reduced by unity is halved and then squared. And the square is then added to its own root (mula), then multiplied by the common difference (attha (asta in Sanskrit, eight) in the verse, d), and then joined with the first term (panca in the verse, five, a) . The result thus obtained when multiplied by the terms (indayas in the verse), yields the sum (pudhavidana prthvidhana in Sanskrit) in the verse, Sm).”

i.e. Sm = [3]

The context of the application of the above formula in the TP needs to be explained here that there are three types of cavaties distributed all over the seven earth : central cavaties, cavaties arranged in rows and scattered cavaties5. Of them the first two form a larger A.P. having d = 8 and a = 5.

It, now, becomes clear that the problem to solve before Yativrsabha was to compute the sum of a group of certain terms in a larger A.P. and, for that, ge deduced a user friendly formula.

From historical point of view the formula [3] is particular and singular, and from padegogical point of view it is fascinating. Therefore, T.A. Sarasvati Amma (c. 1920 A.D. – 2000 A.D.), a great scholar of Indian mathematics rightly states that the formula is a distinct contribution of the TP to series mathematics6.
Sarasvari Amma has given an excellent derivation to the formula. Her efforts made in deriving the same are praiseworthy.7

Here we shall employ the principle of mathematical induction, using her technic, to prove the formula.

Let P (m) : Sm = odd m N.
For m = 1, we have P (1) : S1 = a, clearly P (1) is true.
For m = k, let P(k) be true.
Sk =

Now let us suppose that the first term of the group of (k + 2) terms be the rth term in the larger A.P.

Then r = (1 + 3 + ….. + k) + 1

Let k be the sth term in the A.P. within the brackets with common difference being equal to 2.
Then, on applying the formula [1],

k = 1 + (s – 1) 2

Then, on applying the formula [2],

or

Therefore, the rth term in the larger A.P.

= a + ( 1 — 1 ) d
Therefore,
Sk+2 =
or Sk+2 =
or Sk+2 =

Since we have assumed P (k) to be true, we find P (k + 2) is true.

Hence, it follows by the principle of mathematical induction that P(m) is true for all odd natual numbers.

Now, we shall find out the formula for the case when m is even.

Let us suppose that the first term of the group of m terms be the rth term in the larger A.P.

Then r = ( 2 + 4 + 6 + …….. + ( m – 2 ) + 1

Let (m -2) be the sth term in the A.P. within the brackets with common difference being equal to 2. Then, on applying the formula [1].

m – 2 = 2 + ( s – 1 ) 2

Then, on applying the formula [2].

or
Therefore, the rth term in larger A.P.

= a + (2 – 1 ) d
= a +
Therefore,
Sm
or Sm =

ACKNOWLEDGEMENTS

We are thankful to Dr. Anupam Jain (indore) to encourage us for preparing this paper.

Reference

1.Patni, C.P. [Editor[ Tiloyapannatti of Yativrsabha (with Aryanka Visuddamati’s Hindi Commentary), 3 vols, vol-I, Second Edition, Sri 1008 Candraprabha Digambra Jain Atisayaksetra, Dehra-Tijara, 1997.

2.Sastri, N.C. Jain [1992] Tirthankara MAhavira aura Unaki Acarya PArampara (in Hindi), 3 vols., vol. -I. Acarya Santisagara Chani Granthamala, Mujaffarnagar, p 87. First Edition by Bharatvarsiya Digambara Jaina Vidvatparisada in 1974.

3.Jain, L. C., [2003] Mathematical Contents in the Digambara Jaina Texts of the Karnanuyoga Group, 2 vols., vols.- I Indian National Science Academy’s Project, Published by Kundakunda Jnanapitha, Indore, p. 33.

4.Sarasvati Amma, T.A. p1961-62 “The Mathematics of the First Four Mahadhikaras of the  Trilokaprajnapti ”. 

Journal of the Ganganatha Jha Research Institute, 18, 27-51, esp.pp. 34-41.

5.For details vide : Jadhav “On the Sum”.

6.Sarasvati Amma, T.A. [1961-62] “The First Four”, p. 51.

7.Ibid, pp. 36-38.

अर्हंत वचन, अक्टूबर दिसम्बर २००५