# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३७१

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175
CHAPTER VI--MIXED PROBLEMS.

${\displaystyle {\tfrac {1}{2}}}$, and 7 is the number of terms. If you are acquainted with calculation, then tell me quickly what the sum of the series of fractions in geometrical progression here is.

The rule for arriving at the sum of a series in geometrical progression wherein the terms are either increased or decreased (in a specified manner by a given known quantity):-

314. The sum of the series in (pure) geometrical progression (with the given first term, given common ratio, and the given number of terms, is written down in two positions) ; one (of these sums so written down) is divided by the (given) first term. From the (resulting) quotient, the (given) number of terms is subtracted. The (resulting) remainder is (then) multiplied by the (given) quantity which is to be added to or to be subtracted (from the terms in the proposed series ). The quantity (so arrived at) is (then) divided by the common ratio as diminished by one. (The sum of the series in pure geometrical progression written down in) the other (position) has to be diminished by the (last) resulting quotient quantity, if the given quantity is to be subtracted (from the terms in the series). If, however, it is to be added, (then the sum of the series in geometrical progression written down in the other position) has to be increased by the resulting quotient (already referred to. The result in either case gives the required sum of the specified series)

Examples in illustration thereof.

315. The common ratio is 5, the first term is 2, and the quantity to be added (to the various terms) is 3, and the number of terms is 4. O you who know the secret of calculation, think out and tell me quickly the sum of the series in geometrical progression, wherein the terms are increased (by the specified quantity in the specified manner).

34. Algebraically, ${\displaystyle \pm \left({\tfrac {s}{a}}-n\right)m\div (r-1)+s}$ is the sum of the series of the following form: ${\displaystyle a,ar\pm m,(ar\pm m)r\pm m{\Big \{}(ar\pm m)r+-m{\Big \}}r\pm m}$ and so on,