Summation of Series.
Hereafter we shall expound in (this) chapter on mixed problems the summation of quantities in progressive series.
The rule for arriving at the sum of a series in arithmetical progression, of which the common difference is either positive or negative:-
290. The first term is either decreased or increased by the product of the negative or the positive common difference and the quantity obtained by having the number of terms in the series as diminished by one. (Then, this is (further) multiplied by the number of terms in the series. (Thus, the sum of a series of terms in arithmetical progression with positive or negative common difference is obtained.
Examples in illustration thereof.
290.[1] The first term is 14; the negative common difference is 3; the number of terms is 5. The first term is 2; the positive common difference is 6; and the number of terms is 8. What is the sum of the series in (each of these cases?
The rule for arriving at the first term and the common difference in relation to the sum of a series in arithmetical progression, the common difference whereof is positive or negative :-
292.[2] Divide the (given) sum of the series by the number of terms (therein) and subtract (from the resulting quotient) the product obtained by multiplying the common difference by the half of the number of terms in the series as diminished by one. (Thus) the first term (in the series) is arrived at. The sum of the series is divided by the number of terms (therein). The first term is subtracted (from the resulting quotient); the remainder when divided by half of the number of terms in the series as diminished by one becomes the common difference.
290.^ Algebraioally, , where n is the number of terms, a the first term, b the common difference, and s the sum of the series.
292.^ Algebraically, and .
