316. The first term is 3, the common ratio is 8, the quantity to be subtracted (from the terms) is 2, and the number of terms is 10. O you mathematician, think out and tell me quickly what happens to be here the sum of the series in geometrical progression, whereof the terms are diminished (by the specified quantity in the specified manner).
The rule for arriving at the first term, the common difference and the number of terms, from the mixed sum of the first term, the common difference, the number of terms, and the sum (of a given series in arithmetical progression):-
317.[1] (An optionally chosen number representing) tho number of terms (in the series) is subtracted from the (given) mixed sum. (Then) the sum of the natural numbers (beginning with one and going up to) one less than this optionally chosen number is combined with one. By means of this as the divisor (the remainder from the mixed sum as above obtained is divided). The quotient here happens to be the (required) common difference; and the remainder (in this operation of division ) when divided by the (above optionally chosen) number of terms as increased by one gives rise to the (required) first term.
An example in illustration thereof.
318. It is seen here that the sum (of a series in arithmetical progression) as combined with the first term, the common difference, and the number of terms (therein) is 50. O you who know calculation, give out quickly the first term, the common difference, the number of terms, and the sum of the series (in this case).
The rule for arriving at the common limit of time when one, who is moving (with successive velocities representable) as the terms in an arithmetical progression, and, another moving with steady unchanging velocity, may meet together again (after starting at the same instant of time):-
317.^ See stanzas 80-82 in Ch. II and the note relating to them.
