A Treatije of Algebra. 187

+

,» + 1 1

And, by following the same Me-

thod the Sums of other Series may be determined, Butbrfore I leave this Subject, it may not be amiss to givean Example or two, of the Use of these two last Ex-pressions. Let it be proposed to find the Sum of the Pro-gression 7 2 + 9* + IjZ + 1Z 2 , continu’d to 20Terms: In this Cafe, thescommon Difference (or e) be-ing = 2, m will be = 5 j therefore, by writing 20, 2and 5 instead of n, e and m, in the general Expression

n 1 . 1 . e 1 *

nm z -j- n . n + 1 . me -f-

we

shall

20.21.41.4

have 20.25 -f- 20.21 .10 ---= loiao

6

for the Number sought.

Again, suppose the Sum of the fir st 10 Terms of theProgression 2 + ^2)3 -f 3 -f 2 S /2V -f- 4 + 3V'al 5 +5 -}- 4^/21 3 , were required ; then, 1 being x 4-^/2,m will be = 1; therefore by writing 10, 1, and i + \/ zfor n, m and e respectively, in the last generalF.xpres-

sion, it will become 10 -}-

10 . 11 .r

10 . 11 . 3 ->.i 4- ^2

+

IOO . I2X

r+yTs

2 _ 4

2481s 17600^/2, the Value sought in this Cafe.

If any One be desirous to fee this Speculation carry’dfurther, so as to extend to Series of Powers whose In-dices are Fractions, such as Square-Roots, Cube-Roots,isfc. I must refer him to my EJsays, where it is treatedin a general Manner j and desire him to observe here,once for all, that the Theorems above found, will holdequally, in Cafe of a descending Series, such as m — e z-j- m — 2r] 2 , (3V. or m — #'.3 -j- m — 2 eV , & c . provid-ed the Signs of the second, fourth, fsV. Terms bechanged ; this is evident from the Investigation.

Though