# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२१९

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23
CHAPTER Ī - ARITHMETICAL OPERATIONS.

term and the common difference to 8 times the common difference multiplied by the sum of the series, the common difference is added, and the resulting quantity is halved; and when (again) this is diminished by the first term and then divided by the common difference, we gat the number of terms in the series.

The rule for finding out the number of terms (stated) in another manner:--

70. When, from the square root of (the quantity obtained by) the addition of the square of the difference between twice the frst term and the common difference to 8 times the common difference multiplied by the sum of the series, the kșēpapada is subtracted, and (the resulting quantity) is halved; and (when again this is) divided by the common difference, (we get) the number of terms in the series.

Examples in illustration thereof.

71. The first term is 2, the common difference 8; these two are increased successively by 1 till three (series are so made up). The sums of the three series are 90, 276 and 110 in order. What is the number of terms in each Series ?

72. The first term is 5, the common difference 8, and the sum of the series 333. What is the number of terms ?

The first term (of another series) is 6, the common difference 8, and the sum 420. What is the number of terms?

The rule for finding out the common difference as well as the first term :--

73. The sum (of the series) diminished by the ādidhana,and (then) divided by half (the quantity represented by) the square

70. Kșēpapada is half of the difference between twice the first term and the the common difference. i.e. ${\displaystyle {\frac {2a-b}{2}}}$. It is obvious that this stanza varies the rule mentioned in the previous stanza only to the extent necessitated by the introduction of this kșēpapada therein.

73. For ādidhana and uttaradhana, see note under stanzas 63 and 64 in this chapter. Symbolically expressed this stanza works out thus:--

${\displaystyle b={\frac {S-na}{{\Big \{}{\frac {n^{2}-n}{2}}{\Big \}}}}}$ and ${\displaystyle a={\frac {S-{\frac {n(n-1)}{2}}b}{n}}}$