SECTION XIV. 
THE INVESTIGATION OF THE SUMS OF POWERS OF 
NUMBERS IN ARITHMETICAL PROGRESSION. 
B ESIDES the two sorts of progressions treated of 
in Section 10, there are infinite varieties of other 
kinds; but the most useful, and the best known, are 
those consisting of the powers of numbers in arithmeti 
cal progression ; such as 1* + $ % + 3 Z F 4* .. . . ri 1 , 
and l 3 + 2 3 + 3 3 + 4 z .... ri 3 , &c. where n denotes 
the number of terms to which each progression is to be 
continued. In order to investigate the sum of any such 
progression, which is the design of this Section, it will 
be requisite, first of all, to premise the following 
LEMMA. 
If any expression, or series, as 
An f B// 2 + Cn? -|- D/2 4 See.} ■ , . 
-an- be- c*- <ie &c J.' n '-°lv.ng the powers 
,of an indeterminate quantity n, be universally equal to 
nothing, whatsoever be the value of n; then, I say, 
the sum of the co-efficients A — a, B — b, C — c, See. 
of each rank of homologous terms, or of the same powers 
of n, will also be equal to nothing. 
For, in the first place, let the whole equation 
-an- be - ce &c. t = °- be d,vlded b >’ *’ and 
we shall have i A + l ) a + C "\ f c 'l - o; and 
l a bn + C/2 3 See. S 
this being universally so, be the value of n what 
it will, let, therefore, n be taken = 0, and it will 
A \ _ 
a 5 
0 ; which being rejected, as 
become ^ 
such, out of the last equation, we shall next have 
4- Bn + Cn 2 f D/2 3 &c. £ __ 
— bn — cri — dri 3 See. S ~~ 
again by /?, and proceeding in the very same manner, 
bn — ce — de &c. s - 0 ! wheace ’ dividing