# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२४१

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45
CHAPTER III -- FRACTIONS.

Examples in illustration thereof.

35. In relation to this (given) series, the first term is ${\displaystyle {\tfrac {2}{5}}}$, the common difference is ${\displaystyle {\tfrac {3}{4}}}$ , and the sum given is ${\displaystyle {\tfrac {7}{54}}}$; again (in relation to another series), the common difference is ${\displaystyle {\tfrac {5}{8}}}$, the value of the first term is ${\displaystyle {\tfrac {3}{8}}}$, and the sum is ${\displaystyle {\tfrac {3}{40}}}$. In respect of these two (series), O friend, give out the number of terms quickly.

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

36. The sum (of the series) divided by the number of terms (therein) when diminished by (the product of) the common difference multiplied by the half of the number of terms less by one, (gives) the first term (of the series). 'I'he common difforence is (obtained when) the sum, divided by the number of terms and (then) diminished by the first term, is divided by the half of the number of terms less by one.

Examples in illustration thereof.

37. Give out the first term and the common difference (respectively) in relation to (the two series characterised by) ${\displaystyle {\tfrac {31}{150}}}$ as the sum. and having ${\displaystyle {\tfrac {3}{4}}}$ (in one case) as the common difference and ${\displaystyle {\tfrac {4}{5}}}$ as the number of terms, and (in the other case) ${\displaystyle {\tfrac {1}{3}}}$ as the first term and ${\displaystyle {\tfrac {4}{5}}}$as the number of terms.

The rule for finding out in relation to two (series), the number of terms wherein is optionally chosen, their mutually interchanged first term and the common difference, as also their sums which may be equal, or (one of which may be) twice, thrice, half or one- third (of the other):-

38. The number of terms (in one series) multiplied by itself as lessened by one, and then multiplied by the chosen (ratio between the sums of the two series), and then diminished by twice the number of terms in the other (series, gives rise to the interchangeable) first term (of one of the series). The square of the

36. See note under 74, Chap. II.

38. See note under 86, Chap. II.