Contribution of Mahaviracharya in the

Author(s): R S Lal
Publisher: Z_Deshbhushanji_Maharaj_Abhinandan_Granth_012045.pdf

Page 4

"The number of terms (in one series), multiplied by itself as lessened by one and then multiplied
by the chosen ratio between the sums of the two series, and then diminished by twice the number of terms in the other series gives (rise to the interchangeable) first term of one (of the series). The square of the (number of terms in the) other (series) diminished (again) by the product of two (times the) chosen (ratio) and the number of terms (in the first series) gives (rise to the interchangeable common difference (of that series)."
Symbolically if S, S, be the sums, a, a, the first terms and d, d, the common differences of the two given series then a1 = Sa and did. Now if - r and n, n, be the respective number of terms in the
S
S
two series, then according to the above formula
a=n (n-1) x r-2n,
and
Example1:
and
In relation to two men (whose wealth is measured) respectively by the sums of two series in A.P. having 5 and 8 for the number of terms, the first term and the common difference and both these series be interchangeable (in relation to each other), the sums (of the series) being equal or the sum (of one of them) being twice, thrice, or any such (multiple of that of the other), 'O arithmetician, give out the (value of these) sums and the interchangeable first term and common difference after calculating (them all) well." Solution :
If
S=S, then r=1
so in the above case where n=5 and n,-8 we have
Then
and
which proves that
1.
६६
GSS
d-(n,)-n,-2rn
2
पंचाष्टगच्छपुंसोर्व्यस्तप्रभवोत्तरे समानयनम् । द्विगुणादिधनं वा हि त्वं गणक विगणय्य ॥
a=n (n-1) x 1-20,
=5 (5-1) x 12 x 8 2016 4
d (D1) D1 - 2 rn
-
(8)282 X 1 X 5 64 8 10
r=
-
s=2(2 x 4 (2 x 4+ (5-1) x 46) 16) = 5 (4+92) = 480
8
S-(2 x 46+ (8-1)x4)-4 (9+28) - 480
S =1.
S
27
9 46
87
आचार्य रत्न श्री देशभूषण जी महाराज अभिनन्दन ग्रन्थ