THE SUMS OP PROGRESSIONS. 
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B X n + 1 = I + 2 + 3+4 K + » + 1. 
From which the first equation being subtracted, there re 
mains A x n + l j 1 — A «' + B x n + l — Bnrr n + l: 
this contracted will be 2 A n + A -f B = n + l; whence 
we have 2 A ■— l x n 1- A 1- B — l ~0: wherefore, 
by taking 2A — l = 0, and 4 +B — 1 = 0 (according 
to the lemma) we have A = i, and B = and con 
sequently l + 2 + 3 +.4 . ... ,n ( — An z + 13/2) — 
n z ^ n n x n + 1 * 
Case 2 Ô . To find the sum ofi the progression \ z + 2* + 
3 Z h z , or l + 4+9 + 16.... n\ 
Let A/2 3 +- Bn z +- C«, according to the aforesaid ob 
servations, be assumed = l 2 4- 2 Z + 3 Z +- 4 2 . ... n 2 : 
then,by reasoning as in the preceding case, we shall have 
A. x it + 11 + B x n +- l J -+ C x n + l = l z +- 2 2 + 
3 2 + 4 2 .... id + n +- i( 2 ; that is, by involving n +- l 
to its several powers, An 3 -+ 3A n z + 3A/2 + A + Brd 
+ 2B« + B + Cn + C = l 2 -+ 2 2 A 3 Z -+ 4 2 .... n z 
-+ : from which, subtracting the former equa 
tion, we get 3A>2 2 + 3 A/2 -+ A + 2B/2 + B + C 
( = n -+ lj 1 ) = n z +- 2/2 + 1 ; and consequently 
* In this investigation it is taken for granted, that the 
sum of the progression is capable of being exhibited by 
means of the powers of n, with proper co-efficients : 
which assumption is verified by the process itself; for 
it is evident from thence, that the quantities An z + B//, 
and 1 + 2 + 3 + 4 . . . /2, under the values of A and 
B there determined, are always increased equally, by 
taking the value of n greater by an unit .: if, therefore, 
they are equal to each other, when n is = o (as they 
actually are) they must also be equal when n is 1 ; and 
so likewise, when n is 2, &c. &c. And the same reason-» 
jug holds in all the following cases.