INCREMENTS. 



Here the generul term is (f -}- 1 ) '' ; and the integral of Ex. j — Required the fum of any number n terms of the 



this, that is, /(.r f 1)' =/..' + 3 f^-- + Sf" + / >. ^'"" I . a + 2.3 + 3 .4 + 4- 5 + &'^- « (n + 



is compuud as follows : ^ j^^^.^ ^j^^ g^^^,.^! term of the feries is x (.x- +1), it is 



^ , _ x' _ :l' 4. •'^'" "^ ■'■' therefore required to find the integral of (A-f I ) (.v + 2). 



J 4A.V 2 4 riril by art. 10. wc have 



3/ «• = ffi. - ^r - ^' / <■' -)(•■-> = ^^^^^^ = 



»/".,: — _3j1_ _ 3 -^ — ^ '- — i '- by writing x = n, and S x = l. 



^ The fame may alfo be found, as in the example above, 



f I = by article S ; but it is needlefs to repeat it, as it only dif- 



-^ •'■■ firs from that in its conftant fador. 



We have in neither of the foregoing examples any 

 correction, for it is obvious that when .r = o, the formula 



" ' ^ ^> v'^ + ^ ) _ ^^ ^^ jt ought to be, and there - 



fori the integral needs no correction. 



Hence a very curious property with regard to the fums of ^.v. 6. — Required the fum of any number of terms n, of 



confecutive cubes h«Tinning at I ; -viz. that tliis fum is the feries 



equal to the fquare of'the fum of all their roots. , .4.7.10 + 4.7. 10.13 + 7 . 10 . 13.16 + Sec. 



Thus, i5+2> = (i + ar ^ «{« + 3) (» + 6) (« + 9). 



l' + 2' + 3' — (i + 2 + 3)- j^g^g t]j(. general terra of the feries is .r (.r + 3) {x + 6) 



I' 4- 2' -i- 3' + 4' = (i + 2 + 3 4- 4)' (.,. + ^^, and ^-e have therefore to find the integral of 



^*'- , /""' ■ , ■ • •„ (■»■ + V) (.'• f 6) (.r + 9) (x + 12). Now, by art. 10. 



As to the correftion in thofe cafes that required it, it will , 



be found as in the preceding example; as alfo the fums _/ (.v + 3) (.r + 6) { x + g) (x + 12) = 



of any number of cubes wliofe roots are in arithmetical pro- ^ 



greffion, having any common difference m. And the fums ■ •^- (^ -f 3"* (-'■• + 6) (-'• + 9) (.v + 12) _ 



of any powers whatever are attainable upon the fame prin- J 5 A .v 



ciples. 1 „ (n + 3) (;■■ + 6) jr. + 9) (n + 12) 



£x. 4. — Required the fum of any propofed number of |^ ij 



terms in the feries of triangular numbers ^^_ ,,-r[thg x = h, and A .r = 3. 



' / ■. Hx.'j. — Required the fum of the infinite feries 



In this cafe the general term is ^^— i^ — ' I i i i 



+ - — + , , - + --. + &c. 



And here, fince A .y — I, ^\e have 



Or, by making .v = n, the fum required is ( i 



and it will therefore be necelTary to find the integral of 1-2.3 2.3.4 3.4.5 4-5 



6 



Here the crcncral term of llie fe 



(■^ + ^)l'^+-) = if (.,+ ,) (, + ,) but (art. ,0) «(,:+!) (. f.) " """ ^'" S-cral term ot tr,e ler.es 



= X /■ '.+ t) (x + 2) = -± +^U^ + ^) ^ - 7{7r^h^TT)''"^'^'' moftfimple idea is now 



I 'j_ \ I _u 2) ^° confider the feries as generated from its extremity, 



" \" • ' '^ : 1 by making x = n, and A x = I. which is infinitely diftant ; under which circumil.iiice each 



6 , -r r 1 c 01. term will be the increment of all thofe which follow it in 



The fame may be otherwiie found from art. 8 ; thus, t^e above arrangement ; and therefore, in order to find the 



^ f(x+i) (x + 2) = ij\' +lfs X + J f 2 f„^ beginning at any term _ ' , we have 



„i ,.1 ,. A, ,. •"•' (•"■' + •) (* "i- 2) 



^ J ' 6A.r 4 13 fimply to fir.d the integral of ~ — — 



i r ^ _ _i.— _ 2j1 taken negatively, beeaufe the increment x, that is» 



^ J 4 A A- 4 ;i .V = — I. Now, by art. II, 



J ' - aT J{x- 1} (x) (X + 1} ~ Ax ^ 2x (X + i)~ 



And fince A .x = i, this fum becomes ^ . l^^caufe Ax=^ i; that is, the fum of the 



r' x"- X 2 -^ (■»' + 1 ) 



h fix- + I) {X + 2) =-.-+-+- = ' . c ■ r • u • • 



-' 023 above infinite fenes beginning at any term 



X (^ + I) (^ + 2) And the fums Of any order Of poly- , n(„ + i)(„4-2> 



o is equal to ■ ; and therefore when n = i, the 



gonal and figurate numbers may be found in the fame 2 n (h + i) 



manuer. whole feries becomes = ^. 



Cor. 



