पृष्ठम्:Ganita Sara Sangraha - Sanskrit.djvu/२२३

CHAPTER II-ARITHMETICAL OPERATIONS. The common difference (in respect of another series) is 5, the number of terms is 8, and the sum is 156. Tell me the first term. 25 1 The rule for finding out how (when the sum is given) the first term, the common difference, and the number of terms may, as desired, be arrived at - 78. When the sum is divided by any chosen number, the divisor becomes the number of terms (in the series); when the quotient here is, diminished by any number chosen (again), this subtracted number becomes the first term (in the series); and the remainder (got after this subtraction) when divided by the half of the number of terms lessened by one becomes the cominon difference. Example in illustration thereof. 79. The sum given in this problem is 540. O crest-jewel of arithmeticians, tell me the number of terms, the common differ- ence, and the first term. Three rule-giving stanzas for splitting up (into the component elements such a sum of a series (in arithmetical progression) as is combined with the first term, or with the common difference, or with the number of terms, or with all these. 80 O crest-jewel of calculators, understand that the misradhana diminished by the uttaradhuna, and (then) divided by the number of terms to which one has been added, gives rise to the first term 81. The misradhana, diminished by the addhona, and (then) divided by the (quantity obtained by the) addition of one to the (product of the) number of terms multiplied by the half of the number of terms lessened by one, (gives rise to) the common 78 Symbolically, the problem herein is to find out o. when S is given, and a and are allowed to be chosen at option Naturally, there may be in relation to any given value of Sany values of b, which depend upon the chosen values of a aud . When the values of a andare de limitely chosen, the rule herein given for finding out b turns out to be the same as that given in stanza 74 above. 80-82. The expression misradhana means a mixed sum. It is used here to denote the quantity which may be obtained by adding the first tem or the common difference or the number of terms or all three of these to the sum of a