PROGRESSION. ALGEBRA 537 called simply proportionals. Thus, 12, 4, 15, 5, are four numbers in geometrical proportion; and, in general, na, a, nb, b, may express any four proportionals, for = n, and ct , rib also = n . o To denote that any four quantities a, b, c, d, are pro portionals, it is common to place them thus, a : b : : c : d ; or thus, a : b = c : d; which notation, when expressed in words, is read thus, a is to b as c to d, or the ratio of a to b is equal to the ratio of c to d. The first and third terms of a proportion are called the antecedents, and the second and fourth the consequents. When the two middle terms of a proportion are the same, the remaining terms, and that quantity, constitute three geometrical proportionals; such as 4, 6, 9, and in general na, a, - . In this case the middle qiiantity is called n a mean proportional between the other two. 51. The principal properties of four proportionals are the following : 1. If four quantities be proportionals, the product of the extremes is equal to the product of the means. Let a, b, c, d, be four quantities, such that a : b : : c : d; then, from the nature of proportionals, 7 = -, : let these equal quotients be multiplied by b d, and we have ad = be. It follows, that if any three of four proportionals be given, the remaining one may be found. Thus, let a, b, c, the first three, be given, and let it be required to find x, the fourth term ; because a : b : : c : x, ax = bc, and dividing , bo bya,z= . The converse is obviously true, viz., if four quantities be such that the product of two of them is equal to the pro duct of the other two, these quantities are proportionals. 2. If four quantities are proportional, that is, if a : b : : c : d, then will each of the following combinations or arrange ments of the quantities be also four proportionals. 1st, By inversion, b : a : : d : c . 2d, By alternation, a : c : : b : d . Note. The quantities in the second case must be all of the same kind. 3d, By composition, a + b :a : c + d:c, or, a + b :b : c + d :d. 4:th, By division, a-b :a : c-d :c, or, a-b :b : c-d:d. 5th, By mixing, a + b : a-b :: c + d : c-d. 6tk, By taking any equimultiples of the antecedents, and also any equimultiples of the consequents, na :pb : : nc :pd. 7tk, Or, by taking any parts of the antecedents and con- a b c d sequents, -:-::-:- n p n p That the preceding combinations of the quantities a, b, c, d, are proportionals, may be readily proved, by taking the products of the extremes and means ; for from each of them we derive this conclusion, that ad=bc, which is known to be true, from the original assumption of the quantities. 8th, If four quantities be proportional, and also other four, the product of the corresponding terms will be proportional. Let a : b :;c :d, And e :f ::g :Ji; Then ae : bf : : eg : dh . For ad=bc, and eh=fg, as before, therefore, multiplying together these equal quantities, adeh = bcfg, or aexdh= bf x eg; therefore, by the converse of the first property, ae : Hence it follows, that if there be any number of pro portions whatever, the products of the corresponding terms will still be proportional. 52. If a series of quantities be so related to each other, that the quotient arising from the division of any term by that which precedes it is always the same quantity, these are said to be in geometrical progression; such are the numbers 2, 4, 8, 16, 32, &c., also i, , ^, -j 1 ^, &c., and in general, a series of such quantities may be represented thus, a, ar, ar 2 , ar 3 , ar 4 , ar 5 , &c. Here a is the first term, and r the quotient of any two adjoining terms, which is also called the common ratio. By inspecting this series, we find that it has the follow ing properties : 1. The last term is equal to the first, multiplied by the common ratio raised to a power, the index of which is one less than the number of terms. Therefore, if z denote the last term, and n the number of terms, z = ar n ~ l . 2. The product of the first and last term is equal to the product of any two terms equally distant from them : thus, supposing ar 5 the last term, it is evident that a x ar 5 = arx ar 4 = ar 2 x ar 5 , &c. The sum of n terms of a geometrical series may be found thus : Let Then Subtract, That is, Hence s = a + ar + ar- + ar 3 . . . + ar n rs = ar + ar 2 + ar 3 . . . + ar n ~ l rs s = ar n a . (r-l) 5 = (r-l). r" 1 1 r* s = -a, or a. r -1 1-r a Cor. The sum to infinity = - Additional Examples in Proportion and Progression! Ex. 1. How many strokes does a clock strike in twelve hours ? If s denote the number 5= 1+ 2 + .. .12 . . 2s=13 + 13 + ... 13 = 13x12; s = 78. Ex. 2. Find the number of shot lying close together in the shape of an equilateral triangle. Let n be the number of shot in a side of the triangle. Counting from one angle, and taking in successive rows parallel to the opposite side, we get as the number re quired Ex. 3. To find the number of shot in a pile of the form of a triangular pyramid. As each shot lies in the hollow formed by those below it, the number of shot in the successive sides from the base upwards will evidently be n-l, n-2,...l . Hence the number of shot in the pile will be n(n + l) (n-I)n (re-2)(-l) 1.2 -- - _L. - -i- - - -- 1- -- * 222 2 To sum this series induction may be employed. The 1 result is NX. 4. A ratio of greater inequality is diminished, and of less inequality increased, by adding the same quantity to each of its terms. ,,,7" T , , ,, a + x a Let a > b ; then - <-. b+x b By multiplying out, this is evident.
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