# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/२२२

एतत् पृष्ठम् परिष्कृतम् अस्ति
26
GAŅIITASĀRASAŃGRAHA

difference. (In splitting up the number of terms from the miśradhana), the (required) number of terms (is obtained) in accordance with the rule for obtaining the number of terms, provided that the first term is taken to be increased by one (so as to cause a corresponding increase in all the terms).

82. The miśradhana is diminished by the first term and the number of terms, both (of these) being optionally chosen: (then) that quantity, which is obtained (from this difference) by applying the rule for (splitting up) the uttara-miśradhana happens to be the common difference (required here ). This is the method of work in (splitting up) the all-combined (miśradhana).

Examples in illustration thereof.

83. Forty exceeded by 2, 3, 5 and 10, represents (in order) the ādi-miśradhana and the other (miśradhana). Tell me what (respectively) happens in these cases to be the first term, the common difference, the number of terms and all (these three).

series in arithmetical progression. There are accordingly four different kinds of miśradhana mentioned here; and they are respectively ādi-miśradhana' and uttara-miśradhanas', gaccha-miśradhana and sarva-miśradhana. For ādidhana and uttaradhana sē note under stanzas 63 and 64 in this chapter.

Algebraically, stanza 80 works out thus: $a={\frac {Sa-{\frac {n(n-1)}{2}}b}{n+1}}$ , where Sa is the ādi-miśradhana, i.e.,$S+a$ .

And stanza 81 gives $b={\frac {Sb-na}{{\frac {n(n-1)}{2}}+1}}$ where $Sb$ is the uttara-miśradhana,
i.e., $S+b$ ; and further points out that the vlaue of n may be found out, when the value of Sn, which, being the gaccha-miśradhana, is equal to $S+n$ , is given, from the fact that, when $S==a+(a+b)+(a+2b)+...$ upto n terms.
Since, in stanza 82, the choice of a and n are left to our option, the problem of finding out a,n, and b from the given value of S a n b, which, being the sarva-miśradhana, is equal to $S+a+n+b$ , resolves itself easily to the finding out of b from any given value of Sb in the manner above explained.

83. The problem expressed in plainer terms is:-- (1) Find out a when $Sa==42$ , and $n==5$ . (2) Find out b, when $Sb==43,a==2$ and [/itex]n==5[/itex]. (3) Find out n when $S+n==45,a==2$ and $b==3$ . And (4) find out a,b, and n when $S+a+b+n==50$  