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 pro- gressive 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 multi- plied 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 column of figues so derived and given in the margin- 0 0 the lowest 1 18 multiplied by r, which gives since this lowest 1 has 0 above it, the r obtained as before is squared, which gives since this 0 has 1 above it, the r now obtained is multiplied by r, which gives r3, since this 1 has 0 above it, this is squared, which gives and since. 1 again this 0 has another 0 above it, this is squared, which gives ¹. 1 0 Thus the value of r may be arrived at by using as few times as possible the processes of squating and simple multiplication. The object of the method 18 to facilitate the determination of the value of r", and it is easily seen that the method holds true for all positive and integral values of n. ar 1xr - a r-1 95. Expressed algebraically, S = The antyadhana is the value of the last term in a series in geometrical progression, for the meaning and value of gunadhana, see stanza 93 above in this chapter. The antyadhana of a geometrically progressivc series of n terms 18 a-1, while the gunudhana of the same series is ar". Similarly the antyadhana of a geometrically progressive scries of n 1 terms 18 a"-2, while the junadhana thereof is a/-1 Here it is evident that the antyadhana of the series of terms is the same as the gunadhana of the series of -1 terms.
