# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२३१

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CHAPTER II - ARITHMETICAL OPERATIONS.

these (quantities) by one and (then) halve it and multiply it by the common difference; and (then) add the first term to (each of) these (resulting products). And these (resulting quantities), when multiplied by the remaining number of terms and the chosen-off number of terms (respectively), give rise to the sum of the remainder-series and to the sum of the chosen-off part of the series (in order).

The rule for obtaining in another manner the sum of the remainder-series and also the sum of the chosen-of part of the given series :--

107. (Take) the chosen-off number of terms as combined with the total number of terms (in the series), and (take) also the chosen-off number of terms (simply); diminish (each of) these by one, and (then) multiply by the common difference, and (then) add to (each of) these (resulting products) twice the first term. These (resulting quantities), when multiplied by the half of the remaining number of terms and by the half of the chosen-off number of terms (respectively), give rise to the sum of the remainder-series and to the sum of the chosen-off part of the series (in order).

The rule for finding out the sum of the remainder-series in respect of an arithmetically progressive as well as a geometrically progressive series, as also for finding out the remaining number of terms (belonging to the remainder-series) :--

108. The sum (of the given series) diminished by the sum of the chosen-off part (of the series) gives rise to the sum of the remainder-series in respect of the arithmetically progressive as well as the geometrically progressive series; and when the difference between the total number of terms and the chosen-off number of terms (in the series) is obtained, it becomes the remaining number of terms belonging to that (remainder-series) .

107. Again, ${\displaystyle S_{v}=={\Big \{}(n+d-1)b+2a{\Big \}}}$, and ${\displaystyle S_{i}=={\Big \{}(d-1)b+2a{\Big \}}{\frac {d}{2}}}$.