(number of terms in the) other (series), diminished by that (number of terms) itself, and (then) 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).
Examples in illustration thereof.
39. In relation to two series, having and to (respectively) represent their number of terms, the first term and the common difference are interchangeable, the sum of one (of the series) is either a multiple or a fraction (of that of the other, this multiple or fraction being the result of the multiplication or the division as the case may be) by means of (the natural numbers) commencing with 1. O friend, give out (these) suns, the first terms and the common differences.
The rule for finding out the guņadhana, and the sum of a series in geometrical progression :--
40. The first term (of a series in geometrical progression), when multiplied by that Self-multiplied product of the common ratio, in which (product) the frequency of the occurrence of the common ratio is measured by the number of terms (in the series), gives rise to the guņadhana. And it has to be understood that this ( guņadhana), when diminished by the first term and (then) divided by the common ratio lessened by one, becomes the sum of the series in geometrical progression.
The rule for finding out the last term in a geometrically progressive series as well as the sum of that (series):--
41. The antyadhana or the last term of a series in geometrical progression is the guņadhana 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)
40. See note under 93, Chap. II.
41. See note under 95, Chap. II.
