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.
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 common 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 difference, 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 miśradhana diminished by the uttaradhana, and (then) divided by the number of terms to which one has been added, gives rise to the first term.
81. The miśradhana , diminished by the ādidhana, 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 b, when S is given, and a and m are allowed to be chosen at option. Naturally, there may be in relation to any given value of S any values of b, which depend upon the chosen values of a and n. When the values of a and n are definitely 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 miśradhanameans a mixed sum. It is used here to denote the quantity which may be obtained by adding the first term or the common difference or the number of terms or all three of these to the sum of a
