(6) The first term of a decreasing arithmetical series is 10, the common difference, and the number of terms 21; required the sum of the series. Ans. 140. (7) One hundred stones being placed on the ground in a straight line, at the distance of 2 yards from each other; how far will a person travel who shall bring them one by one to a basket which is placed 2 yards from the first stone? 5.4 Ans. 11 miles and 840 yards. The relations (1) and (2), in which five quantities, a, d, n, l, S, enter, will serve to determine any two of these when the other three are given. Thus they furnish the solution of as many distinct problems as there are ways of taking two quantities from among five; and, consequently, the number of problems will be For 10. In order that they may be possible, it is necessary that the value of n should be not only real, but entire and positive. Without entering into the details of the calculation, we place below the solutions of these ten problems. བྱཱ 0? 229. A series of quantities, in which each is derived from that which immediately precedes it, by multiplication by a constant quantity, is called a Geometrical Progression, or Progression by Quotients. Thus, the numbers 2, 4, 8, 16, 32, .... in which each is derived from the preceding by multiplying it by 2, form what is called an increasing geometrical progression; and the numbers 243, 81, 27, 9, 3, in which each is derived ... from the preceding by multiplying it by the number decreasing geometrical progression. The common multiplier in a geometrical progression is called the common ratio. Generally, if a be the first term and p the common ratio, the successive terms of the series will be of the form The exponent of p in the second term is 1, in the third term is 2, in the fourth term 3, and so on; hence the nth term of a series will be of the form, αρα-1, 230. To find the sum of n terms of a series in geometrical progression. Multiply both sides of the equation by p, Sp= ap+ap3+ap3+............... ·+ap"1+ap". If the series be a decreasing one, and consequently p fractional, it will be convenient to change the signs of both numerator and denominator in the above expressions, which then become 231. If two progressions have different first terms, but the same ratio, the ratio of the sums of the two is equal to the ratio of their first terms. For (a+ap+ap2+ap3+, &c.): (b+bp+bp2+bp3+, &c.) a(1+p+ p2+ p3+, &c.): b(1+p+ p2 + p3+, &c.)=a:b 232. It appears that if any three of the five quantities, a, l, p, n, S, be given, the remaining two may be found by eliminating between equations (1) and (2). It must be remarked, however, that when it is required to find p from a, n, S given, or from n, 1, S given, we shall obtain p in an equation of the nth degree, a general solution of which can not be given. If n be required, it will be convenient to apply logarithms, as the equation to be resolved will be an exponential. EXAMPLE I. Required the sum of 10 terms of the series 1, 2, 4, 8, .... a=1, p=2, n=10 To insert m geometric means between a and b. Here we are required to form a geometric series, of which the first and last terms, a and b, are given, and the number of terms =m+2; in order, then, to determine the series, we must find the common ratio. or -1 m+1 ... ita. m-1 a+m+ab+m+1am-1b2+...+m+1a2bm-1+m+ab+b, a+a+1bm+1+a+1bTM+1+...+am+1bm+1+am+1bm+1+b. 233. To find the sum of an infinite series decreasing in geometrical progression. We have already found that the sum of n terms of a decreasing geometrical series is Since p is a fraction, p" is less than unity, and the greater the number n, the smaller will be the quantity p"; if, therefore, we take a very great number of αρη terms of a decreasing series, the quantity p", and, consequently, the term 1 -p' a will be very small in comparison with ; and if we take n greater than any assignable number, or make n=∞, then p" will be smaller than any assignable number, and therefore may be considered =0, and the second term in the above expression will vanish. Hence we may conclude that the sum of an infinite series, decreasing in geometrical progression, is a Strictly speaking, is the limit to which the sum of any number of terms approaches, and the above expression will approach more or less nearly to perfect accuracy, according as the number of terms is greater or smaller. Thus, let it be required to find the sum of the infinite series The error which we should commit in taking for the sum of the first n 2 terms of the above series is determined by the quantity as the sum of 5 terms of the above series, the amount 2 1 would be too great by 162° 3 1 If we take as the sum of 6 terms, the amount will be too great by 486' and so on.* I. The theory of progressions involves that of logarithms. Let there be two progressions, the one geometric, beginning with 1, the other arithmetical, beginning with 0. 1:2:4:8:16:32:64:128, &c. which exhibit a notation sometimes employed. If we compare these with each other, we perceive that, multiplying together any two terms of the first, and adding the corresponding terms of the second, we obtain two corresponding terms, again, of these same progressions. Thus, 4×16=64, 6+12=18; and we perceive that 18 corresponds to 64. Thus a multiplication is effected by addition. This simple observation is, no doubt, very ancient; but it was the genius of Napier, a Scottish baronet, which derived from it the theory of logarithms, one of the most useful of modern discoveries. It was published in 1644, under the title of Mirifici Logarithmorum Descriptio. Logarithms, then, according to Napier, were regarded as a series of numbers in arithmetical progression, while the numbers themselves corresponding, formed a geometrical progression. I proceed to explain his method of constructing them. In order that the geometrical progression should embrace all numbers greater than 1, it is necessary to conceive it formed of terms which increase in an insensible manner, setting out from 1; and, to have their logarithms, it is necessary to conceive the arithmetical progression as composed of terms which vary by insensible degrees, setting out from zero. At their origin, the simultaneous increments which the terms 1 and 0 receive are inappreciably small; but, however small they may be, we may conceive that there is a certain relation established between them, which is entirely arbitrary. Thus, when these increments begin to arise, we can suppose that that of the logarithm 0 is double, triple, &c., of that of the number 1. This relation is called the modulus of the logarithms, which desig nate by M. Suppose, now, that to the term 1 of the geometric progression an increment w, very small, but yet appreciable in numbers, is given. The corresponding increment of the term zero of the arithmetical progression will be very nearly equal to Mo; and we can take for the two progressions these: 1:1+w:(1+w)2: (1+w)3: (1+w)1: &c. 20. Μω. 2Μω . 3Μω . 4Μω . &c. We have said that the relation or modulus M can be taken at pleasure; consequently, according to the values attributed to it, will be obtained different systems of logarithms. The logarithms which Napier published were derived from the progressions which supposes M=1. 1:1+w:(1+w)2: (1+w)3: &c. -0. ω. 2w. 36 &c., This avoids the multiplications by M. The logarithms of numbers in Napier's table serve to find those of any other system, by simply multiplying each by the modulus of that system. The terms of these two series vary slowly, so that, in prolonging both as far as we please, we are sure of finding in the first, terms equal to the entire numbers 2, 3, &c., or so near them that the difference may be neglected. The corresponding terms of the second may then be taken for the logarithms of these numbers, and are those written in the tables. By this we perceive that these logarithms are not exactly those of the numbers beside which they are written. But there is another cause of inaccuracy, viz., that w represents only approximately the increment, which the logarithm 0 takes when w is that taken by 1. The smaller is, however, the greater the exactness. II. Let it be proposed to determine the error produced by assuming that the difference of the numbers is proportional to the difference of their logarithms, when the number of places in the numbers is 5, and their difference not greater than 1. If in the series [A], Art. 224, we make n=, 1 we have |