# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२४४

एतत् पृष्ठम् परिष्कृतम् अस्ति
48
GAŅITASĀRASAÑGRAHA.

Examples an illustration thereof.

44. T'he number of terms are 5, 4 and 3 (respectively) and the common ratios as well as the (equal) common differences are ${\displaystyle {\tfrac {4}{3}},{\tfrac {5}{3}}}$ and ${\displaystyle {\tfrac {7}{3}}}$ (in order). What is the value of the (corresponding) first terms in relation to these (sets of two series, one in geometrical progression and the other in arithmetical progression), which are oharacterised by sums of the same value ?

Thus ends the summation of fractions in series,

Vyutkalita of fractions in series.

The rule for performing the operation of youtkrc:lita is as follows :--

45. (Take) the chosen-of number of torms as combined with the total number of terms (in the series), and (take) also your chosen-off number of terms (separately). Multiply each of theso quantities by the common difference and diminish (the products ) by the common difference; (then) multiply by two; and these (resulting quantities), when multiplied by the half of the remaining number of terms and by the half of the chosen-off number of terms (respectively), give rise to the sum of the remainder-series and to the sum of the chosen-off part of the (given) series (in order).

The rule for inding out the first term in relation to the remaining number of terms (making up the remainder-series) :--

46. The first term (of the series), diminished by the half of the common difference, and combined with the chosen-off number of terms as multiplied by the common difference, as also with the half of the common difference, (gives) the first term of the remaining number of terms (making up the remainder-series). And the common difference (of the remainder-series) is the same as what is found in the given series.

45. Cf. note under 106, Chap. II.

46. Cf. note under 109, Chap. II.