CHAPTER III-FRACTIONS. 45 Examples in illustration thereof. 35. In relation to this (given) series, the first term is , the common difference is , and the sum given is ; again (in relation to another series), the common difference is, the value of the first term is, and the sum is In respect of these two (scries), O friend, give out the number of terms quickly The rule for finding out the first term as well as the common difference :- 36 The sum (of the series) divided by the number of terms (therein), when diminished by (the product of) the common difference multiplied by the half of the number of terms less by one, (gives) the first term (of the series) The common difference is (obtained when) the sum, divided by the number of terms and (then) diminished by the first term, is divided by the half of the number of terms less by one. Examples in illustration thereof. 37. Give out the first term and the common difference (res- pectively) in relation to (the two series characterised by) as the sum, and having (in one case) as the common difference and as the number of terms, and (in the other case) as the first term and as the number of terms. The rule for finding out in relation to two (series), the number of terms wherein is optionally chosen, their mutually interchanged first term and the common difference, as also their sums which may be equal, or (one of which may be) twice, thrice, half or one- third (of the other) :- 38. The number of terms (in one series), multiplied by itself as lessened by one, and then multiplied by the chosen (ratio between the sums of the two series), and then diminished by twice the number of terms in the other (series, gives rise to the inter- changeable) first term (of one of the series). The square of the 36. See note under 71, Chap. II. 38. See note under 86, Chap. II.
