terms quantities term equal proportionals series
SECT. V.-PROPORTION AND PROGRESSION.
In comparing together any two quantities of the same kind in respect of magnitude, we may consider how much the one is greater than the other, or else how many times the one contains either the whole or some part of the other; or, which is the same thing, we may consider either what is the difference between tire quantities, or what is the quotient arising from the division of the one quantity by the other: the former of these is called their arithmetical ratio, and the latter their geometrical ratio. These denominations, however, have been assumed arbitrarily, and have little or no connection with the relations they are intended to express.
I. Arithmetical Proportion and Progression.
a, a + d, a +2d, a + 3d, a + 4d, a +5d, a + Gd, &c., where a denotes the first term, and d the common difference.
By a little attention to this series, we readily discover that it has the following properties: The last term of the series is equal to the first term, together with the common difference taken as often as there are terms after the first. Thus, when the number of terms is 7, the last term is a + 6d; and so on. Hence if z denote the last term, n the number of terms, and a and d express the first term and common difference, we have z = a + (n - 1)d.
The sum of the first and last term is equal to the sum of any two terms at the same distance from them. Thus, suppose the number of terms to be 7, then the last term is a + 6d, and the sum of the first and last 2a + 6d; but the same is also the sum of the second and last but one, of the third and last but two, and so on till we come to the middle term, which, because it is equally distant from the extremes, must be added to itself.
To find the sum of the series, it is only necessary to observe that, if the progression is written down twice, 1° from the beginning, 2° from the end, the terms of the former increase by the same amount as that by which the terms of the latter diminish; so that the sum of any two terms which stand under each other is always the same, viz., the same as the sum of the first and last terms; hence the double series converts addition into multiplication; so that if s denote the sum of the series, we have 2s = n(a +z), and s =(a+z) .
Es. The sum of the odd numbers 1, 3, 5, 7, 9, &c., con- tinued to n terms, is equal to the square of the number of terms. For in this case a =1, d= 2, z =1 (n - 1) d - 1, therefore s= 2 - x 2n = n2.
II. Geometrical Proportion and Progression.
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 Lag= n, and also nb b =n To denote that any four quantities a, b, c, d, are proportionals, 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 arc the same, the remaining terms, and that quantity, constitute three geometrical proportionals; such as 4, 6, 9, and in general na, a, a - . In this case the middle quantity is called a mean proportional between the other two.
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, b -= -: 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 by a, x= be The converse is obviously true, viz., if four quantities be such that the product of two of them is equal to the product of the other two, these quantities are proportionals.
If four quantities are proportional, that is, if a : b c : d, then will each of the following combinations or arrangements of the quantities be also four proportionals.
1st, By inversion, b : a:: d : c .
2d, By alternation, a : c : : b :d .
..Vote. - 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.
4th, 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.
6th, By taking any equimultiples of the antecedents, and also any equimultiples of the consequents, na :pb ::ne :pd .
sequents, - : : - : - • That the preceding combinations of the quantities a, 5, 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 Then ae :bf cg :dh For ad= be, and eh = fg, as before, therefore, multiplying together these equal quantities, adeh = bcfg, or ae x dh = bf x cg; therefore, by the converse of the first property, ae bf : : cg : dh .
Hence it follows, that if there be any number of proportions whatever, the products of the corresponding terms will still be proportional.
By inspecting this series, we find that it has the follow- incf° properties: The last term is equal to the first, multiplied by the common ratio raised to a power, the index of which is ono less than the number of terms. Therefore, if z denote the last term, and n the number of terms, z=ar*-1.
The product of the first and last term is equal to the product of any two terms equally distant from them : thus, supposing ar5 the last term, it is evident that a x ar5 = ar x ar4 = ar2 x ar3, &c.
The sum of n terms of a geometrical series may be found thus : Cor. The sum to infinit a y = 1 - r' Additional Examples in Proportion and Progression.' Ex. 1. How many strokes does a clock strike in twelve hours?
If s denote the number s= 1+ 2 + ...12 2s=13+13+...13=13 x 12; 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 required • 2 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-1, n- 2,...1 Hence the number of shot in the pile will be n(n+ 1) (n -1)n (n -2)(n-1) 1.2 To sum this series induction may be employee. Tho result is ,n(n+1)(n+2) Ex. 4. A ratio of greater inequality is diminished, and of less inequality increased, by adding the same quantity to each of its terms. ..7: Let a> b ; then - b+x+< By multiplying out, this is evident.
Ex. 5. Find the vulgar fraction which is equivalent to the recurring decimal.
Ex. 6. A sum of money doubles itself in fifteen years at a rate a little below 5 per cent. A noble Scotch family have retained in their possession gold coins of the valise of £500 since the days of Mary Stuart (300 years) ; what have they lost by not allowing the money to accumulate at the above rate ?
Every pound would have amounted to £220; .% 524,000,000.
Ex. 7. The sum of the mixed series