पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३६८

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172
GAŅITASĀRASAŃGRAHA.

difference between the first term and the common difference (in the series). (Then) the square of the sum (of the series) is multiplied by the common difference. If the first term is smaller than the common difference, then (the first of the products obtained above is) subtracted (from the second product). If, however, (the first term is) greater (than the common difference), then (the first product above-mentioned is added (to the second product). (Thus) the (required) sum of the cubes is obtained.

Examples in illustration thereof.

304. What may be the sum of the cubes when the first term is 3, the common difference 2, and the number of terms 5; or, when the first term is 5, the common difference 7, and the number of terms 6 ?

The rule for arriving at the sum of (a number of terms in a series wherein the terms themselves are successively) the sums of the natural numbers (from 1 up to a specified limit, these limiting numbers being the terms in the given series in arithmetical progression):-

305–305${\displaystyle {\tfrac {1}{2}}}$.[1] Twice the number of terms (in the given series in arithmetical progression) is diminished by one and (then) multiplied by the square of the common difference. This product is divided by six and increased by half of the common difference and (also) by the product of the first term and the common difference. The sum (so obtained) is multiplied by the number of terms as diminished by one and then increased by the product obtained by multiplying the first term as increased by one by the first term itself. The quantity (so resulting) when multiplied by half the number of terms (in the given series) gives rise to the required sum of the series wherein the terms themselves are sums (of specified series)

305-305${\displaystyle {\tfrac {1}{2}}}$.^  Algebraically, ${\displaystyle \left[\left\{{\tfrac {(2n-1)^{2}}{6}}+{\tfrac {b}{2}}+ab\right\}(n-1)+a(a+1)\right]{\tfrac {n}{2}}}$ is the sum of the series in arithmetical progression, wherein each term represents the sum of a series of natural numbers up to a limiting number, which is itself a member in a series in arithmetical progression.