182 A Treatise of Algebra.'

let, therefore, x be taken = o, and it will become

A

>—a

}

= o, or A—s — o; which being rejected, as such,out of the last Equation, we shall next have

I - - S* - % C C ] } = °; whence by divid-

ing again by *, and proceeding in the very same Man-ner, B — b is also proved to be = o ; and from thence,C~c, D — d, tic. tic. QT. D.

Now to apply what is here laid down to the Purpofoabove specified, it will be proper to observe, first, that,as the Value of any Progression, (i 2 -j- a 2 -f $ 2 -j- 4*.n 2 , or i 3 r? -j- 3.3 -j- 4.3 , . ,, . « 3 ) varies ac-cording as (n) the Number of Terms varies, it must,if it be possible to be expressed in a general Manner, beexplicable by n and its Powers with determinate Co-efficients j secondly, it is obvious that these Powers, inthe Cafes above proposed, .must be rational,, or suchwhose Indices are whole positive Numbers, because theProgression, beirig an Aggregate of whole Numbers, can-not be universally expounded by Surds; lastly, it willappear that the greatest of those Indices cannot exceedthe common Index of the Progression by more' thanUnity ; for, otherwise when n is taken indefinitely great,the highest Power of n would be indefinitely greater thanall the rest of the Terms of the Equation put together.Thus, the highest Power of n, in an Equation univer-sally exhibiting the Value of i 2 -(- 2 2 -j- 3 1 .» 2 ,

cannot be greater than ; for i 2 -J- 2 2 -f- 3* ., . . » 2is manifestly less than n 3 (or n % -j- n 2 -f- n 2 -f- tic. con-tinued to n Terms) but when n is indefinitely great,is, it is plain, indefinitely greater than » 3 , or any otherinferior Power of n, and therefore cannot enter into theEquation. This being premised, the Method of Investi-gation may be as follows:

Case i c . Tofind the Sum of the Prognffion 1 -f z -f 3 -f 4 • • •. n.

Let A« 2 -(- Brc be assumed, according to the forego-ing Observations, as an universal Expression for 1 -{- 2

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