# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२१६

एतत् पृष्ठम् परिष्कृतम् अस्ति
20
GAŅIITASĀRASAŃGRAHA.

57. Give the cube roots of 270087225844 and 76332940488. 58. Give the cube roots of 77308776 and also of 269917119. 59. Give the cube roots of 2427715584 and of 1626379776. 60. O arithmetician, who are clever in calculation, give out after examination the root of 859011369945948864, which is a cubic quantity.

Thus ends cube root, the sixth of the operations known as Parikarman.

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Summation.

The rule of work in relation to the operation of summation of series, which is the seventh (among the perikunnan operations), is as follows:-

61. The number of terms in the series is (first) diminished by one and (is then) halved and multiplied by the common difference; this when combined with the first term in the series and (then) multiplied by the number of terms (therein) becomes the sum of all (the terms in the series in arithmetical progression)

The rule for obtaining the sum of tho series in another manner:- 62. The number of terms (in the series) as diminished by one and (then) multiplied by the common difference is combined with twice the first term in the series; and when this (combined sum) is multiplied by the number of terms (in the series) and is (then) divided by two, it becomes the sum of the series in all cases,

61. This rule comes out thus when expressed algebraically:-
${\displaystyle \left({\dfrac {n-1}{2}}b+a\right)n==S}$, where a is the first term, b the common difference,n the number of terms and S the sum of the whoele series.
62. Similarly, ${\displaystyle {\dfrac {{\Big \{}(n-1)b+2a{\Big \}}n}{2}}==S}$