PROORESSION. A series of numbers following any law should be called a progression, but the word is usually restricted to two sorts of progreesion, which are called, but by no means correctly, arithmetical and geometrical : the analogies pointed out in Rsoresote give the origin of these terms.
An arithmetical progression Is one in which the terms continually increase or diminish equally, including, as an extreme case, that in which they do not increase or ditniuish at all Thus arc acts of terms in arithmetical progression. The following proposi tion contains the principal part of their theory :— If a be the first term of an arithmetical progression, and Cia the difference between any two terms (negative, if the terms diminish); and if a, be the nth term from and after a exclusive, and the sum of a terms, we have From these two equations between a, n, a., Aa, and s, , any three of these being given, the other two can be found, subject however to this restriction, that the problem is unmeaning when n is not a whole number, whether it be given or found. These theorems are only the simplest ease of a more general pair, in which, taking any series, and supposing neither the differences nor the differences of the differences, eta, to he equal, an expression is given for any term of a aeries, or for the sum of n terms, which frequently gives definite forms in the place of indefinite ones. Calculate, as in the article DIFFERENCE, the value of Aa, Ara, from a, a„ a„ &c., and let Thus in the series 1 + 5 + 17 + 43 + 89 + 161 + kcs, the law of whose terms is undiscoverable at first sight, we shall, by what the beginner may, till he knows better, call an accidental circumstance, discover both the law of the terms and that of their sum, as follows:— Thus the seventh term (the sixth after I, n= 6) is or 265, and the sum of 6 terms (make n=6 in the second formula, in which remember that le is the sum of n terms, not of n terms after a) is or 316, which may easily be versified. [Sear.] The apparently accidental circumstance above rdiuded to, is the vanishing of all the differences of a from and after the fourth. But it
is to be observed, that the series was originally constructed so as to make all differences vanish after the fourth, and that the preceding theorem will never change indefinite into definite formulie, except when all differences after a certain one venial. The rule is, when a. is en algebraically rational and integral function of is of the p order, that is, of the form mine 4. , all differences after the pth vanish, and then only.
Geometrical Progression is when the terms of a series increase or diminish by the use of the same multiplier, whole or fractional, including, as an extreme case, that in which the multiplier is unity.
Thus, the multipliers being 1, f, I, and 2, the four following sets of terms are in geometrical progression : 7, 7, 7, 7, &c. 7, V, &c.
g, &c. 9, 18, 36, 72, &e.
If a be the first term, and 5 the second, the nth term is a and the sum of ,s terms is a" —b" b" b) or a)according as a is greater than or less than b. But when a= b, the sum is of course na. If b=a=r, that is, if the terms be a, ar', arc, &c., the nth term is and the sum of terms is 1 —r" r•-1 a — or a 1 — r according as r is less than or greater than one. If r be less than one, the series of terms, however far it may be carried, never reaches a (1— r), though it may approach this limit with any degree of near ness by making the number of terms sufficiently great.
Thus the following equations, 2=1+4 though always erroneous, stop where we may, yet can be brought as near to truth as we please by writing down terms enough on the second side. 1u the use of the word INFINITE, as explained in the article on that subject, we may then say that the above equations are absolutely true, if the series be carried ad infinitum. The general equation, made absolutely true, after stopping at co-" in the series, is a = a + ar + . . . + az.* + Other points connected with this equation will be mentioned in the article SERIES.