*86 A Treatise of Algebra.

But if the Root ot the first Term be not divisible bythe common Difference, as in th* Progression 5* -j- yt-j- 9 l , &c. the Speculation is a little more difficult -nevertheless, the Sum of the Series, in any such Casemay also be found from the same Theorems.

Let the Series m -j- e\ z -f- »*-}■ z*\ z + m -f- 3/ 1 .. .,m -j- be proposed, where m and e denote any Quan-tities whatever, and where n represents the Number ofTerms. Then, by actually involving each Term toits second Power, and placing the Terms in order, thegiven Expression will stand

m 2 -j~ tn 2 - j- m

* m 2 -tn 2 -\-

v. e 2 X-aJ 2 -f-cii

xnme

thus -j zme+\me-\-t>me .... znme ? Now it is evident

' e 2 \-$e 2 —9^* .... n 2 e 2 'that the Sum of the first Rank or Series is n y. tn 2 : Alsothe Sum of the second or 2 mt xi -j- r -s- z -s- 4.., . . »

. , . , n x n 4- 1

appears, by Cafe 1, to be equal to 2 mt x-

2

and that of the third, or e 2 x 1 -j- 4- + 9 + t<f • . . , n z

».«+ I .28+1

(by Cafe 2.) = t 2 x -----: Therefore the

o

: Therefore the

Sum of the wh ole Pro gression m 4 -1 z -j- m -J- 2*1* -f-m-r 3*1* . . . . m-\- net is — n.m i + n . n-f- 1 . me -\-n . n -f- 1 . 2« -f 1 . t 2

6

In like Manner, if the Serie s proposed be

m 4" e 3 "4* 3 -s- m -j- 3«'l 3 . . . . m -s- »el 3 j then

may it be resolved

s 1 + 1 + 1.ixw 3

' I -}- 8 -(- 27 . . . . . hi X <c 3 Jfrom the forementioned Theorems, will appear to be».»-(- 1 . $m 2 e n . n -j- t . 2 n 1 . mt 2

n . m 3 4

2

2