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Shri Mahavir Jain Aradhana Kendra
www.kobatirth.org
Acharya Shri Kailassagarsuri Gyanmandir
CHAPTER II --ARITHMETICAL OPERATIONS.
31 of) this operation) is diminished by one, and (is then) multiplied by the first term, and (is then) divided by the common ratio lessened by one, it becomes the sum of the series).
The rule for finding out the last term in a geometrically progressive series as also the sum of that (series) : .
95. The antyadhana or the last term of a series in geometrical progression is the gunadhana (of another series) wherein the number of terms is less by one. This (antyadhana), when multiplied by the common ratio, and (then) diminished by the first term, and (then) divided by the common ratio lessened by one, gives rise to the sum (of the series).
An example in illustration thereof. 96. Having (first) obtained 2 golden coins (in some city), a man goes on from city to city, earning (everywhere) three times (of what he earned immediately before). Say how much he will make in the eighth city.
Now, in the representative colamn of figures so derived and given in the margin
o the lowest 1 is multiplied by r, which gives r: since this lowest 1 has 0 O above it, the r obtained as before is squared, which gives 12: since this 0
has 1 above it, the qui now obtained is multiplied by r, which gives 78 ; since this 1 has 0 above it, this 18 is squared, which gives p®: and since
again this 0 has another O above it, this is squared, which gives 2
Thus the value of r may be arrived at by using as few times as possible the processes of squaring and simple multiplication. The object of the method is to facilitate the determination of the value of g; and it is easily seen that the method holds true for all positive and integral values of n. 95. Expressed algebraically, S =
S arn 1xr - a
9 -4. The antyadhana is the
value of the last term in a series in geometrical progression; for the meaning and value of guradhana, see stanza 93 above in this chapter. The antyadhana of a geometrically progressive series of n terms is ari-1, while the gunadhana of the same series is arn. Similarly the antyad hana of a geometrically progressive series of n-1 terms is ar -2, while the gunadhana thereof is arn-1, Here it is evident that the antyadhana of the series of » termy is the same 48 the gunadhana of the series of n l terms.
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