. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature. Encyclopedias and dictionaries. INCREMENTS. "by fiiewing the application of the foregoing theory a few examples. Application of the Method of Increments. 14. The fummation of feries, by the inverfe method of increments, is founded generally oii this principle ; that if we have any fefies of quantities, as rt, b, c, d, e, &c. â which are derived from each other according to fome known law; each term may be confidered as the increment ©f tiie fum of all thofe which precede it. Thus, let a-]-l> + c-{-d+e=i
. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature. Encyclopedias and dictionaries. INCREMENTS. "by fiiewing the application of the foregoing theory a few examples. Application of the Method of Increments. 14. The fummation of feries, by the inverfe method of increments, is founded generally oii this principle ; that if we have any fefies of quantities, as rt, b, c, d, e, &c. â which are derived from each other according to fome known law; each term may be confidered as the increment ©f tiie fum of all thofe which precede it. Thus, let a-]-l> + c-{-d+e=i% and a + l> + c + d-\-e+ fâ z': then it is obvious, that 2' â z = ^2 â/*» ^"^ therefore, converfely, the integral of any one of thofe terms, con- fidered as an increment, will reprefent the fum of all the preceding part of the feries. This being premifcd, we may proceed to the folution of the following examples. Ex. I. Required the fum of n tenns of the natural feries i + 2 + 3+4 + Here, by writing x in (lead of «, the term next in order will be .X -)- I, which being the increment of the feries, we (hall have j\x + I) == the fum required. Novv/'(x + 1) =/'^ +y^5 andbyart. 8. Ex. 2. âFind the fum of the feries of fquares i' + f ir S' + r + &c- «'⢠Here the general term of the feries is {x + 2)% and there- fore the furii required will be exprefled by the integral of {x + 2)\ Now /'(x + ^yâ f x"- -f f^x-^ ^4; and by article 8. And fince in this cafe A j = 2, we have r, \-L "^ â¢*â ' 2« Ax^ 4 .r AX = 6+7 A = 1(1 /" I = ⢠2 Aa? 2 And fince in this cafe A ;r = i, we have X X X x"' -f- (x+i)=- â - +- = â:r~ n- -{â n ./ X ⢠/ 2 ' 2 I by writing again « inftead of x, which is the fum of « terms of the propofed feries, as is alfo evident from other conli- derations. ^ .â r, ⢠r i. .. \^\^ example offers an eafy illullration ot what has been obferved at art. 9, of the correftion of an mte-
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