पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२२६

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     The rule for finding out the guņadhana and the sum of a series in geometrical progression :--

93. The first term (of a series in geometrical progression), when 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 number of terms (in the series), gives rise to the guņadhana. And it has to be understood that this guņadhana, when diminished by the first term, and (then) divided by the common ratio lessened by one, becomes the sum of the series in geometrical progression.

    Another rule also for finding out the sum of a series in geometrical progression:--

94. The number of terms in the series is caused to be marked (in a separate column) by zero and by one (respectively) corresponding to the even (value) which is halved and to the uneven (value from which one is subtracted till by continuing these processes zero is ultimately reached); then this (representative series made up of zero and one is used in order from the last one therein, so that this one multiplied by the common ratio is again ) multiplied by the common ratio (wherever one happens to be the denoting item), and multiplied so as to obtain the square (whereever zero happens to be the denoting item). When (the result

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93. The guņadhana of a series of n terms in geometrical progression corresponds in value to the (n + 1)th term thereof, when the series is continued. The value of this guņadhana algebraically stated is r x r x r ..... up to n such factors x a i.e., arⁿ. Compare this with the uttaradhana.

This rule for finding out the sum may be algebraically expressed thus:-- {{space|em|7}$S=={\frac {ar^{n}-a}{r-1}}$, where a is the first term. r is the common ration, and n is the number of terms.

94. This rule differs from the previous one in so far as it gives a new method for finding out rⁿ by using the processes of squaring and ordinary multiplication; and this method will become clear from the following example:--

          Let rⁿ be equal to 12 .

$12$ is even; it has therefore to be divided by 2, and to be denoted by 0;
${\frac {12}{2}}==6$ is even; it has therefore to be divided by 2, and to be denoted by 0;
${\frac {6}{2}}==3$ is odd; 1 is therefore to be subtracted from it, and it is denoted by 1;
$3-1==2$ is even; it has therefore to be divided by 2, and to be denoted by 0;
${\frac {2}{2}}==1$ is odd; 1 is therefore to be subtracted from it, and it is denoted by 1;
$1-1==0$, which concludes this part of the operation.