पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२१७

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CHAPTER II - ARITHMETICAL OPERATIONS.

Tho rule for finding out the ādidhana, the uttaradhanaand the sarvadhana :--

63. Tho ādidhana is the first term multiplied by the number of terms (in the series). The uttaradhana is (the product of) the number of terms multiplied by the common difference (and again) multiplied by the half of the number of terms less by one. The sum of these two (gives) the sarvadhana i.e., the sum of all the terms in the series; and (this sum will be the same as that of a series which is) characterised by a negative common difference, when (the order of the terms in the series is reversed, so that) the last term is made to be the first term.

The rule for finding the 'antyadhana, the madhyadhana and the sarvadhana:-

64. The number of terms (in the series) lessened by one and multiplied by the common differece and (then) combined with the first term (gives) the antyodhana. Half of the sum of

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63-34. In these rules, each of the terms in an arithmetically progressive series is supposed to be obtained by adding to the first term thereof a multiple of the common difference, the nature of this multiple being determined by the position which any specified term holds in the series. According to this conception we have to find in every term of the series the first term along with a multiple of the common difference. The sum of all such first terms so found is what is here called the ādidhana; the sum of all such multiples of the common difference constitutes the uttaradhana; and the sarvadhana which is obtained by adding these two sums is of course the sum of the whole series. The expression antyadhana denotes the value of the last term in an arithmetically progressive series. And madhyadhana means the value of the middle term which value, however corresponds to the arithmetical mean of the first and the last terms in the series, so that when there are 2n + 1 terms in the series, the value of the (n + 1)th term is the madhyadhana, but when there are 2n terms in the series the arithmetical mean of the value of the nth term and of that of the (n+1)th term becomes the madhyadhana. Accordingly we have

(1) Ādidhana $==n*a$.

(2) Uttaradhana == ${\frac {n-1}{2}}*n*b$.

(3) Antyadhana$==(n-1)*b+a$.

(4) Madhyadhana == ${\frac {{\Big \{}(n-1)b+a{\Big \}}}{2}}$

(5) Sarvadhana === $(1)+(2)==(n*a)+({\frac {n-1}{2}}*n*b)$;

or $==(4)*n=n*{\frac {{\Big \{}(n-1)b+a{\Big \}}+a}{2}}$