duct of any two terms, equally distant from the extremes. Let a be the first term, y the last, r the common ratio ; then the series is, a, ar, ar?, ar3, art, &c. y: , , 2, &c. j, 3 It is obvious that any term in the upper rank is equally distant from the beginning, as that below it from the end ; and the product of any two such is equal to ar, the product of the first and last. Prop. II. The fum of a geometrical series wanting the first term, is equal to the sum of all but the last term, multiplied by the common ratio. For, assuming the preceding notation of a series, it is plain that artara tar}, &c. ++++y= =rxa+ar+ar*, &c.. +++ + + Cor. Cor. i. Therefore s being the sum of the series, s-yxr=sa. And s = yra Hence s can be found from a, y, and r; and any three of the four being given, the fourth may be found. Since the exponent of r in any term is equal to the number of terms preceding it; hence, in the last term, its exponent will be n- I ; the last term, therefore Cor. 2. a and sa -I yrar. AX x. Hence of of these four, s, a, r, n, any three being given, the fourth may be found by the solution of equations. If n is not a small number, the cases of this problem will be most conveniently solved by logarithms ; and of such solutions there are exainples in the Appendix to this part. Cor. 3. If the series decreafes, and the number of terms is infinite, then according to this notation, a, the least term, will be o, yr and is ri' Ex. Required the sum of the series 1, Therefore sa 2. 2-I 10 100 What are called in arithmetic, repeating and circulating decimals, are truly geometrical decreasing serieses, and therefore may be summed by this rule. 3 Thus.333,&c. = 3+ + &c. is a geometrical series in which y and r=10, therefore sa yr 3 8 y= 33 24 24 X 100 and r=100, therefore sa = ミ 3хто -1 10 X IOI 100 100 X 100-I It was observed, (Chap. I. and IV.) that in many cases, if the division and evolution of compound quantities be actually performed, the quotients and roots can only be expressed by the series of terms, which may be continued ad infinitum. By comparing a few of the firft terms, the law of the progression of such a series will frequently be discovered, by which, it continued without any further operation. When this cannot be done, the work is much facilitated by several methods; the chief of which is that by the binomial theo formed, ! may be rem. Theorem. Any binomial (as a+6) may be raised to any power (m) by the following rules. 1. From inspecting a table of the powers of a binomial obtained by multiplication, it appears that the terms, without their coefficients are a , a b, a b*, a 63, &c. m M-I m-2 m3 2. The coefficients of these termis will be found by the following rule. Divide Divide the exponent of a in any term by the exponent of b increased by 1, and the quotient multiplied by the coefficient of that term will give the coefficient of the next following term. This rule is found, upon trial in the table of powers, to hold universally. The coefficient of the first term is always 1; and by applying the general rule now proposed, the coefficients of the terins in order will be &c. They may be more conveniently expressed thus, I, Am, BX Сх 3 m m 2 m2- -3 DX 4 &c. the capitals denoting the m 2 2 m 3 preceding coefficient. Hence a+b"=a" + Ama"-'6+Bx"-'xa"-262+CX 3 & b, &c. This is the celebrated binomial theorein. It is deduced here by induction only, but it may be rigidly demonstrated, though upon principles which do not belong to this place. Cor. |