cf Geometrical Series. 331 



r^ the common ratio, and », the number of 

 terms J and fin ce there are only two conditions, 

 given by the problem, whence thefe three quan- 

 tities are to be found, it hath hitherto been 

 thought indeterminate, or one that would ad- 

 mit infinitely many anfwers. 



In an infinite geometrical feries, whofe fum, 

 x-SfXr^xr"^ -\-xr'^ i &c. ad infinitum =:^, is a 

 finite quantity, it is evident that r, the common 

 ratio, mud be lefs than unity : but it may be 

 either a pofitive or a negative quantity. If it 

 be a pofitive quantity, then it mull be a proper 

 fraction j but if negative, it may be either a 

 proper fradlion, or an improper fraftion, or a 

 v/hole number. If r be pofitive, the feries will 

 be, Ar + xr+xr^-{-^r% &c. ad infinitum; but if 

 negative, it will be x—xrj^»'r'^ — ;rr% &c. ad 

 infinitum. I lliall firft folve the general cafe 

 \N\\txt xJrxr-Vxr'^ jfxr'^ y &c. ad infinit. = ^, and 

 x^^x'^r'- ^x^r^^-x-r\ &c. ad infinit. = ^. 



If the firft of thefe equations be multiplied 

 by r, it will be xr\xr'^ ■\xr'^ ^xr'' , &c. ad 

 inf. =^;'j and this, taken from the firft, leaves 

 x — a — ar. 



If the fecond equation be multiplied by r*, it 

 will be;v^r*-j-;^ V^4^ V, &c. ad inf.=^r, and 

 this, taken from ^*4.;fV* +;cV*,&c. ad inf. =^, 

 leaves x^=^ — ^r*. 



But 



