# 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.

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

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