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 guņadhana (of another series) wherein the number of arms 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 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 column of figures so derived and given in the margin--
The table's caption 0 the lowest 1 is multiplied by r, which gives r: since this lowest 1 has 0 0 above it, the r obtained as before is squared, which gives r2 ; since this 0 1 has 1 above it, the r2 now obtained is multiplied by r,which gives r2; 0 since this 1 has 0 above it, this r2 is squared, which gives r2 1 again this 0 has another 0 above it, this r6 is squared, which gives r12
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 rn; and it is easily seen that the method holds true for all positive and integral values of n.
95. Expressed algebraically, , The antyadhana is the value of the last term in a series in geometrical progression; for the meaning and value of guņadhana, see stanza 93 above in this chapter. The antyadhana of a geometrically progressive series of n terms is arn-1, while the guņadhana of the same series is arn. Similarly the antyadhana of a geometrically progressive serires of n-1 terms is arn-2, while the guņadhana thereof is arn-1. Here it is evident that the antyadhana of the series of n terms is the same as the guņadhana of the series of n-1 terms.
