202 
THE INVESTIGATION OF 
B — b is also proved to be, r: 0; and from thence 
C — c , D — d, &c. Q. E. D. 
Now, to apply what is here demonstrated to the pur 
pose above specified, it will be proper tl* observe, first, 
that, as the value of any progression (l 1 + £ f 3 2 
+ 4* ?i l ) varies according as («) the number of 
its terms varies, it must, if it can be expressed in a ge 
neral manner) be explicable by n and its powers with 
determinate co-ellicients; secondly, it is obvious that 
those powers, in the cases above proposed, must be ra 
tional, or such avhose indices are whole positive num 
bers ; because the progression, being an aggregate of 
whole numbers, cannot admit of surd quantities; lastly, 
it will appear that the greatest of the said indices can 
not exceed the common index of the progress ion by more 
than unity; for, otherwise, when n is taken indefinitely 
great, the highest power of n would be indefinitely 
greater than all the rest of the terms put together. 
Thus, the highest power of.«, in an expression univer 
sally exhibiting the value of l 1 d- £ 2 4- 3* 
cannot be greater than n 3 ; for \ z + 2 1 f 3 1 »* 
is manifestly less than n 3 (or id + id -l- a 1 f Sec. con 
tinued to n terms); but « 4 , when n is indefinitely great 
is indefinitely greater than or any other inferior 
power of «,and therefore cannot enter into the equation. 
This being premised, the method of investigation may 
be as follows. 
Case l . To find the sum of the progression 1 + 2 1- 3 
•1 4 .... it. 
Let \rd l B« be assumed according to the foregoing 
observations, as an universal expression for the value 
ol 1 f -j [ 3 .+ 4 // ; where A and B represent 
unknown, but determinate quantities. Therefore,since 
the equation is supposed to hold universally, Avhatsoever 
is the number of terms, it is evident, that, if the num 
ber ot terms be increased by unity, or, which is the same 
thing, it n l- l be wrote therein, instead of //, the equa 
lity will still subsist, ami we shall have A / n I if V