पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२४५

एतत् पृष्ठम् परिष्कृतम् अस्ति
CHAPTER III--FRACTIONS.

47. Even in respect of a geometrically progressive series, the common ratio and the first term are exactly alike (in the given series and in the chosen-off part thereof). There is (however) this difference here in respect of (the first term among the remaining number of terms (constituting the remaindor-series), viz., that the first term of the (given) series multiplied by that self-multiplied product of the common ratio, in which (product) the frequency of occurrence of the common ratio is measured by the chosen-off number of terms, gives rise to the first term (of the remainder-series).

Examples in illustration thereof.

48. Calculate what the sum of the remainder-series is in relation to that (series) of which ${\displaystyle {\tfrac {1}{4}}}$ is the common difference, ${\displaystyle {\tfrac {1}{2}}}$ the first term, and ${\displaystyle {\tfrac {3}{4}}}$ is (taken to be) the number of terms, when the chosen-off number of terms (to be removed) is (taken as) ${\displaystyle {\tfrac {1}{4}}}$.

49. In relation to a series in arithmetical progression, the first term is ${\displaystyle {\tfrac {1}{2}}}$, the common difference is ${\displaystyle {\tfrac {1}{5}}}$, and the number of terms is (taken to be) ${\displaystyle {\tfrac {2}{3}}}$. When the chosen-off number of terms (to be removed) is (taken as) ${\displaystyle {\tfrac {5}{8}}}$, give out, O you who know calculation, the sum of the remainder-series.

50. What is the value of the sum of the remainder-series in relation to a series of which the first term is ${\displaystyle {\tfrac {1}{4}}}$, the common difference is ${\displaystyle {\tfrac {1}{5}}}$, and the number of terms is (taken to be) ${\displaystyle {\tfrac {3}{5}}}$, when the chosen-off number of terms is ${\displaystyle {\tfrac {1}{10}}}$ ?

51. The first term is ${\displaystyle {\tfrac {2}{3}}}$, the common difference is ${\displaystyle {\tfrac {1}{5}}}$, and the number of tenums is (talken as) ${\displaystyle {\tfrac {3}{4}}}$, and the chosen-off number of terms is taken to be ${\displaystyle {\tfrac {1}{5}},{\tfrac {1}{6}}}$ or ${\displaystyle {\tfrac {1}{7}}}$. O you, who, being the abode of kalās * , are the moon shining with the moon-light of wisdom, tell me the sum of the remaining number of terns .

52. Calculate the sum of the remaining number of terms in relation to a series of which the number of terms is 12, the common difference is minus ${\displaystyle {\tfrac {1}{4}}}$, and the first term is ${\displaystyle 4{\tfrac {1}{2}}}$, the chosen-off number of terms being 3, 4, 5 or 8.

7. See note under 110, Chap. II.
* Kāla is here used in the double sense of 'learning' and 'the digits of the moon'