of Ceometrical Series, 222 



the finite feries will be — or ; 



I — r I — r I — r 



and the fum of the fquares will be 



1 — r' 



x'^r'^" x'^ — x'^ r"^" . ^x — xr" „„j , __ 



or : I.e. a = , and e — 



I — r* i-^r"^ i^—r 



X — x^ r 



a n 



J and if x be taken for the leaft term, 

 I — r' 



and xr" for the greateft, in which cafe r will 



be greater than unity, then = a, and 



r — I 



= ^. From thefe two equations 



r^~i 



only, it is impoffible to obtain the values of the 



three unknown quantities, x, r, and n -, recourfe 



muft therefore be had to another property of the 



feries. 



Let there be given x+xr+xr'^+xr^, &c 



to xr^—^^a; a.nd x"" ^x'-r^ -{■x'^r^y &c to 



xr^"—^=lf, where x, r, and n are whole poficive 



numbers. Then the latter of thefe equations 



divided by the former, gives x — xr+xr'^ — 



xr^^-xr* — xrK &c. = — . If this be added to 



the firft equation, the fum will be, ^x-\r2xr~ 

 + 2^r*, &c.=^+ ; and, if it be fubtrafted, 

 the difference will be, '2xr-{-2xr^ +2xr^, Scc.-rz 

 a . Again, let 2x + 2xr'- -i-2xr* + 2xr^y &c, 



= a-\ — be divided by 2xr + 2xr'^ + 2xr^ , &c. 



L ... 



= a , the quotient will give r, and the re- 

 mainder 



