Arithmetic, Rational and Practical Volume 3; Wherein the Properties of Numbers Are Clearly Pointed Out, the Theory of the Science Deduced from First Principles, the Methods of Operation Demonstratively Explained, and the Whole Reduced to Practice, in A G (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1772 Excerpt: ...or series, is formed from some number assumed as the first term, by continual multiplication or division; and is either increasing or decreasing. Thus, 1: 2: 4: 8: 16: 32: 64, is a geometrical progression or series, formed from 1 assumed as the first term, and increasing, being continually multiplied by 2; and may be continued upward to infinity. And 243: 8 1: 27: 9: 3: 1, is a geometrical progression or series, formed from 243 assumed as the first term, and decreasing, being continually divided by 3, and may likewise be continued downward to infinity. The multiplier or divisor whereby the series is continued upward or downward, is called the common ratio. Note. The natural numbers, 1, 2, 3, 4, &c. are sometimes set over a geometrical series, to shew the distance of any term from unity, or from the first term; and in this cafe the natural numbers are called indices or exponents; thus, I. 2. 3. 4. 5. 6. exponents. " 3: 6: 12: 24: 48: 96 series. But if the series proceed from unity, it is usual and convenient to place o over 1, thus, o. 1. 2. 3. 4. 5. 6. exponents 1: 2: 4: 8: 16: 32: 64 series. In a geometrical progression, or series, five things occur to be considered 5 any three of which being given, the other two may be found. The five things are, I least term, ? Extremes II. The greatest term.) III. The number of terms. IV. The common ratio. V. The sum of all the terms. The more useful properties of numbers in geometrical progression are explained in the following theorems. THEOREM I. Any term of a geometrical series is equal to the product of the least term, multiplied continually into the common ratio, repeated as often as there are terms before or after it. Thus in the increasing series, 3: 6: 12: 24: 48: 96, whose common ratio is 2, the term 9...