# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३७०

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174
GAŅITASĀRASAŃGRAHA.

is added (thereunto). The (resulting) quantity when multiplied by the square of the number of terms as increased by the number of terms gives rise to the (required) collective sum.

Examples in illustration thereof.

310${\displaystyle {\tfrac {1}{2}}}$. What would be the (required) collective sum in relation to the (various) series represented by (each of) 49, 66, 13, 14, and 25?

The rule for arriving at the sum of a series of fractions in geometrical progression:-

311${\displaystyle {\tfrac {1}{2}}}$.[1] The number of terms (in the series) is caused to be marked (in a separate column) by zero and by one (respectively), corresponding to the even (value) which is halved, and to the uneven (value from which one is subtracted, till by continuing these processes zero is ultimately reached); then this (representative series made up of zero and one is used in order from the last one therein, so that this one multiplied by the common ratio is again) multiplied by the common ratio (wherever one happens to be the denoting item), and multiplied so as to obtain the square (wherever zero happens to be the denoting item). The result (of this operation) is written down in two positions. (In one of them, what happens to be) the numerator in the result (thus obtained) is divided (by the result itself; then) one is subtracted (from it); the (resulting) quantity is multiplied by the first term (in the series) and (then) by (the quantity placed in) the other (of the two positions noted above). The product (so obtained), when divided by one as diminished by the common ratio, gives rise to the required sum of the series.

Examples in illustration thereof.

312${\displaystyle {\tfrac {1}{2}}}$-313. In relation to 5 cities, (the first term is) dīnāra and the common ratio is ${\displaystyle {\tfrac {1}{2}}}$. (Find out the sum of tho dīnāras obtained in all of them.) The first term is ${\displaystyle {\tfrac {1}{2}}}$, the common ratio is

311${\displaystyle {\tfrac {1}{2}}}$.^  In this rule, the numerator of the fractional common ratio is taken to be always 1. See stanza 94, Ch. II and the note thereunder.