# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२३७

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41
CHAPTER III--FRACTIONS

20. Give the cube roots of ${\displaystyle {\tfrac {125}{8}}}$ and ${\displaystyle {\tfrac {729}{8}}}$

21. O friend of prominent intelligence, give the cube roots of the cubed quantities found in (the examples on) the cubing of fractions and (give also the cube root) of ${\displaystyle {\tfrac {2197}{27}}}$.

Thus end the squaring, square-root, cubing and cube-root of fractions.

Summation of fractional series in progression.

In regard to the summation of fractional series, the rule of work is as follows:--

22. The optional number of terms (making up the fractional series in arithmetical progression) is multiplied by the common difference, and (then it) is combined with twice the first term and diminished by the common difference. And when this (resulting quantity) is multiplied by the half of the number of terms, it gives rise to the sum in relation to a fractional series (in arithmetical progression).

Examples in illustration thereof.

23. Tell me what the sum is (in relation to a series) of which ${\displaystyle {\tfrac {2}{3}},{\tfrac {1}{6}}}$ and ${\displaystyle {\tfrac {3}{4}}}$ are the first term, the common difference and the number of terms (in order); as also in relation to another of which ${\displaystyle {\tfrac {2}{5}},{\tfrac {3}{4}}}$ and ${\displaystyle {\tfrac {2}{3}}}$(constitute these elements).

24. The first term, the common difference and the number of terms are ${\displaystyle {\tfrac {3}{5}},{\tfrac {1}{5}}}$ and ${\displaystyle {\tfrac {2}{3}}}$ (in order in relation to a given series in arithmetical progression). The numerators and denominators of all (these fractional quantities) are (successively) increased by 2 and 3 (respectively) until seven ( series are so made up). What is the sum (of each of these) ?

22. Algebraically ${\displaystyle (nb+2a-b){\tfrac {n}{2}}}$. Cf. note under 62, Chap. II

23. Whenever the number of terms in a series is given as a fraction, as here, it is evident that such a series cannot generally be formed actually number of terms. But the intention seems to be to show that the rule holds good even in such cases.