fourth; for if the unknown term be an extreme, it will be found by subtracting the other extreme from the sum of the means; and if it be one of the means, by subtracting the other mean from the sum of the ex Then { b + c = + x and (4 + 8 = x + 3 ⋅ b + c = = x [4+8-3= x (15) It hence follows also, that if two terms, as a and b, are given, a third arithmetical proportional to them may be easily found, in order to form an arithmetical progression. For if we suppose the required term to be x, we shall have: Consequently, to find a third arithmetical proportional to two given terms, we must subtract the first from double the second. Thus, the third arithmetical proportional to 3 and 7, will be 14 -311; and indeed 3. 7. 11. An arithmetical mean proportional between two given terms, such as a and b, may be found with equal ease; for if the required mean be denoted by a, we shall have Which indicates, that an arithmetical mean proportional to two quantities, is equal to the half of these quantities. Thus, the mean proportional between 9 and 13, is 11; for 9. 11. 13. THEOREM II. 22. In every even arithmetical progression, the sum of all the terms, equally distant from the extremes, taken two and two, is equal to that of the extremes; and if it be odd, the sum of the extremes, or of any two terms equally distant from the extremes, is the double of the mean term. In the first case, the sum of an arithmetical progression, is equal to the product of the sum of the extremes multiplied by half the number of terms; and in the second, to the product of the mean multiplied by the number of terms. THEOREM III. 23. In every arithmetical progression, any term whatever is equal to the first and as many times the common difference as there are terms before it. It hence follows, that we may easily find the value of any term of an arithmetical progression, the first term, the common difference, and the number of terms of which are known. For example, the 121st term of an arithmetical progression, the first term of which is 5 and the common difference 3, will be 365; for 5 + 3 x 120 = 365. Properties of Geometrical Proportion and Progression. THEOREM I. 24. In every geometrical proportion, the product of the extremes is equal to that of the means. If 3 6 4 8 Then 3 x 8 = 6 x 4 Consequently, the fourth term of a geometrical proportion, the other three of which are known, may be easily found; for if the required term be an extreme, it will be equal to the product of the means divided by the other extreme; and if it be a mean, it will be equal to the product of the extremes divided by the other mean. Two terms being given, a third, geometrically proportional to them, may be easily found, in order to form a geometrical progression. Let us suppose that a third proportional is required to the terms a and b, and that the term sought is denoted by y. We shall then have 18 Consequently, to find a third term, geometrically proportional, we must divide the square of the second, or its product by itself, by the first term. Thus the third geometrical proportional to 3 and 6, will be 6 x 6 = 12; and indeed ÷ 3: 6:12. 3 A mean geometrical proportional between two terms, as a and b, may be found with equal ease; for if this term be called r, we shall have: Thus, if we suppose a 2, and b equal to the square root of 16, which is 4. 2:4:8. 8; x will be And indeed THEOREM II. 25. Any term whatever of a geometrical progression, is equal to the product of the first term multiplied by the common ratio, raised to that power the exponent of which is equal to the number of terms before it. Let the geometrical progression be 2:4:8: 16 : 32: &c. The fifth term 32, for example, is equal to the product of 2, the first, multiplied by 16, which is the fourth power of the ratio 2. THEOREM III. 26. In every geometrical progression, the second term, less the first, is to the first, as the last, less the first, is to the sum of all those which precede it. 4:8:16:32: &c. If ÷ 2 RULE OF THREE. 27. The Rule of Three, is an operation by which, when three terms of a geometrical proportion are known, a fourth, not known, may be found; and it is called direct when the similar terms increase in the same ratio. For example: if four men perform six yards of work in a certain time, it is evident that a greater number must perform more in the same time. On the other hand, if the similar terms, instead of increasing in the same ratio, must decrease, the rule is called inverse, as is the case in the following example: If four men perform a certain work in eight days, a greater number of men must perform it in a time proportionally less. The Rule of Three Direct, and the Rule of Three Inverse, may be expressed by the following formulæ. Men Men Yards Yards Days Days 3 : 6 :: x : 8 The Rule of Three is compound or simple, according as the terms are compound or simple. For example, the above two formulæ express each the simple rule of three. But if it were required to divide the profits of a commercial company among several partners, who have advanced certain capitals, for different periods of time, it would be necessary to multiply the capital of each partner by the time, which would render the rule the compound rule of three. As the rule of three is only the application of the formulæ of Theorem I. (23.) it is needless to enlarge farther on this subject. |