SERIES. 



From this general formula are readily drawn the following particular expreffions for the fums of the different orders 

 of polygonal and figurate numbers ; as alfo for the fquares, cubes, and higher powers. 



Figurste Numbers. 



Series. General term. Sum of n terms. 



« (n + I) 

 1 + 2+3+4+ " = I . 2 



7i (n + i) « (n + l) (n + 2) 



I + 3 + 6 + ,0 + -^-y- = — rr^-r^ 



« (n + I) (« + 2) _ n (« + l) {n + 2) 

 I + 4 + 10 + 20 + 1.2.3 ~ 1.2.3 



1 + 5 + 15 + 35 + ^''- = ^'=' 



where the law of continuation is fufficiently obvious. 



Polygonal Numbers. 



Series. General term. Sum of n terms. 



n (n- l) 

 1 + 2+3+4+ .... n = «-l- ^ ^ 



n' + n 2n (n— l) n (n— l) (n— 2) 

 I + 3 + 6 + 10 + .... — — = n + —7-7^- + 1.2.3 



2n'' — on 3 » (n — I ) 2 n (« — l) (n — 2) 



i-r'f-ryr I 1.2 1.2 1.2.3 



3B^ — n 4«(n— l) 3n(«— i)(b— 2) 



, + 5 + 12 + 22 + .... -— - = „ + —TTr~ + 1.2.3 — 



, , , . (m — 2) n^ — (m — 4) n 

 univerfally ; the general term bemg — -— 



( w- I) »(«- I) (m-2)« («- l) («-2) 



The fum of n terms = n 4- -— — ■ — H ; 



1 . z 1 . z . 3 



Series. General term. Sum. 



«' n' n 

 i^ + 2* + 3^ + 4^ + «= = 7 + T ^ "6 



n* n' n' 



1^ + a' 4- 3' + 4' + «' = :r + T + 7 



424 



n' n' n' n 



I' + 2» + 3* + 4' + «' = — + — + 



^ 523 30 



I' + 2^ + 3^ + 4^ + «^ = 4. 4 ^ _ _ 



2 12 12 



&C. &C. &C. = &C. 



A variety of other feries fall under the above general The following formulae, all relating to the differential 



formula of M. Montmort ; -viz. feries of which the fum method, will not be unacceptable to the reader, 



may be exhibited in a finite form : and in all cafes where Let a + b -^c-^il-^e + f+ &c. be any feries ; 



the fuccefTive differences decreafe, an approximation may be make 



obtained by it, and that with a confiderable degree of fa- D' = i — a 



cihty, when the terms are alternately 4- and — i, but -qv — ^ ~ 2b + c 



when they are all plus, or all minus, except the firft, little, j)iii =: a — ib + ac — d 



if any, advantage is gained by it. D'' = a — Ai + 6c — ±d + e 



The above method of fummation is commonly called the 

 J(^f«n/w/w!rf;Jo(/,and wasiirllufedfori«/cr/iote/onbyBriggs, 



in the conftruftion of his table of logarithms. Newton alfo £)(")_ a — nb + " ^"~ '' c — '"~ ^' v'~^J 4. &c. 



applied it to a variety of interefting problems in his " Me- 1.2 1.2.3 



thodus DifFerentialis ;" but Montmort, as far as we have been From which laft general formula the firft term of any order 



able to trace, was the firft who employed it in the fumma- of differences may be found independent of all thofe which 



tion of feries. precede it. 



6 Again, 



