Elements of Algebra

164 ELEMENTS SECT. III. compounded of the number of workmen and that of the days which they have been employed. If, for example, 25 sous per day be given to one mason, and it is required what must be paid to 24 masons who have worked for 50 days, we state the calculation thus: $1 : 24$ $1 : 50$ $1 : 1200 :: 25 : 30000$ sous, or 1500 francs. In these examples, five things being given, the rule which serves to resolve them is called, in books of arithmetic, The Rule of Five, or Double Rule of Three. CHAP. XI. *Of Geometrical Progressions.* 505. A series of numbers, which are always becoming a certain number of times greater, or less, is called a *geometrical progression*, because each term is constantly to the following one in the same geometrical ratio: and the number which expresses how many times each term is greater than the preceding, is called the *exponent*, or *ratio*. Thus, when the first term is $1$ and the exponent, or ratio, is $2$, the geometrical progression becomes, Terms $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ &c. Prog. $1, 2, 4, 8, 16, 32, 64, 128, 256, \&c.$ The numbers $1, 2, 3, \&c.$ always marking the place which each term holds in the progression. 506. If we suppose, in general, the first term to be $a$, and the ratio $b$, we have the following geometrical progression: $1, 2, 3, 4, 5, 6, 7, 8 \ldots n.$ Prog. $a, ab, ab^2, ab^3, ab^4, ab^5, ab^6, ab^7 \ldots ab^{n-1}.$ So that, when this progression consists of $n$ terms, the last term is $ab^{n-1}$. We must, however, remark here, that if the ratio $b$ be greater than unity, the terms increase continually; if $b = 1$, the terms are all equal; lastly, if $b$ be less than $1$, or a fraction, the terms continually decrease. Thus, when $a = 1$, and $b = \frac{1}{2}$, we have this geometrical progression: