The rule for finding out the first term in relation to the remaining number of terms (belonging to the remainder-series):--
109. The chosen-off number of terms multiplied by the common difference and (then) combined with the first term (of the given series) gives rise to the first term in relation to the remaining terms (belonging to the remainder-series) The already mentioned common difference is the common difference in relation to these (remaining terms also) ; and in relation to the chosen-off number of terms (also both the first term and the common difference) are exactly those (which are found in the given series).
The rule for finding out the first term in relation to the remaining number of terms belonging to the remainder-series in a geometrically progressive series:--
110. Even in respect of a geometrically progressive series, the common ratio and the first term are exactly alike (in the given series and in the chosen-off part thereof). There is (however) this difference here in respect of (the first term in relation to) the remaining number of terms (in the remainder-Series) viz., that the first term of the (given) series multiplied by that self-multiplied product of the common ratio, in which (product) the frequency of the occurrence of the common ratio is measured by the chosen-off number of terms, gives rise to the first term (of the remainder-series).
Examples in illustration thereof.
111. Calculate what the sums of the remainder-series are in respect of a series in arithmetical progression, the first term of which is 2, the common difference is 3, and the number of terms is 14, when the chosen-off numbers of the terms are 7, 8, 9, 6 and 5 (respectively).
112. (In connection with a series in arithmetical progression) here (given), the first term is 6, the common difference is 8, the number of terms is 36, and the chosen-of numbers of terms are 10,
109. The first term of the remainder series . The series dealt with in this rule is obviously in arithmetical progression.
10. The first term of remainder series is .