________________
In the following sutras the rules have been given to find out the sum of a series in geometrical progression, wherein the terms are either increased or decreased in a specified manner by a given known quantity).
गणचितिरन्यादिहता विपदाधिकहीनसंगुणा भक्ता । व्येकगुणेनान्या फलरहिता होनेऽधिके तु फलयुक्ता।।
Algebraically, if S = sum of the series, a = first term
n=number of terms, r = common ratio and m = the quantity to be added or subtracted from each term of the series in G. P., and S' = the sum of the series in G. P., then S - sum of the resulting series
- + S'
r-
1
Proof : Theorem : Let
S-a + (ar + m) + [(ar $ m)r m + ... to n terms and S'= a + ar + ar2 + ar3 + ... to n terms
Now
S =la + ar + arl + ar3 + ... to n terms]
+ m[(r+p2 + ... to n-1 terms)] + m[(r + pa + 3 + ... to n-2 terms)] + ... + m
- 2 = 1 +m">=1+may
+ .. + m
i
-1-1) + (p-2-1) + (pm-3-1) + ... + (r--1)]
+
+
mir(r+re+r+-+pm-1) – 1 (n-1)] -s+ , [ *} - (n-1)]
+ [ 1-n+1]
+
+
1. GSS
172
3 14
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