of Geometrical Series. 335 



Example III, Let a— 1820, and ^= 1660750; 

 then ~= 912I, and « + — =27321, and 4~— = 

 907I, and 27324: divided by 907 i, gives 3 in 

 the quotient, and 10 in the remainder, thus 

 9071)2732^(3 



27 2 



■) T 



10 



fo that 3 is the comnfion ratio, and 5 the firft 

 term; and the feries is 5, 15, 45, 135, 405, 

 1215. 



Example IV. Let ^ = 75, ^ = 2125 j then — 

 = 281 and <?+— = i03f divided by a — ^— 46 1, 

 gives 2 in the quotient, and 10 in the remainder; 

 fo that the feries is 5, 10, 20, 40. 



It remains that fomething be faid of the feries 

 x—xr+xr'^—xr^y &c. ad infinit. Here r is necya- 

 tive ; and fince -37-= the fum of the feries x-\-xr 

 ^xr'^^xr^i &c. ad infinit. confequently, by 

 changing the fign of the quantity r, i. e, by 



writing inftead of the fum of the fe- 



° I +1 1 — r 



• X 



ries x^xr-i-xr' —xr^ i &c. ad inf. will be ■ 



If in this feries r— i, then the feries \% x-~x-i-x 



— x+x, &c. ad infinitum, and its fum=: — - — =: 



i + i 



I have 



