V. If we multiply or divide the two members of an inequation by a negative quantity, the resulting inequation will hold in a sense opposite to that of the original inequation. Thus, if we take the inequation 87, multiplying both members by — 3, we have the opposite inequation, -24 — — 21. VI. We cannot change the signs of both members of an inequation, unless we reverse the sense of the inequation, for this transformation is manifestly the same thing as multiplying both members by — 1. VII. If both members of an inequation be positive numbers, we can raise them to any power without altering the sense of the inequation. That is, if ab then a"b". But, Thus from 2 VIII. If both members of an inequation be not positive numbers, we cannot de termine, a priori, the sense in which the resulting inequation will hold good, unless the power to which they are raised be of an uneven degree. IX. We can extract any root of both members of an inequation without altering the sense of the inequation. That is, if If the root be of an even degree both members of the inequation must necessarily be positive, otherwise we should be obliged to introduce imaginary quantities, which cannot be compared with each other. PROGRESSIONS. ARITHMETICAL PROGRESSION. 178. WHEN a series of quantities continually increase or decrease by the addition or subtraction of the same quantity, the quantities are said to be in Arithmetical Progression. Thus the numbers 1, 3, 5, 7, which differ from each other by the addition of 2 to each successive term, form what is called an increasing arithmetical progression, and the numbers 100, 97, 94, 91, - which differ from each other by the subtraction of 3 from each successive term, form what is called a decreasing arithmetical progression. Generally, if a be the first term of an arithmetical progression, and the common difference, the successive terms of the series will be a, a + d, a ±2d, a ± 3 d, in which the positive or negative sign will be employed, according as the series is an increasing or decreasing progression. Since the coefficient of d in the second term is 1, in the third term 2, in the fourth term 3, and so on, the nth term of the series will be of the form a+ (n-1) d. In what follows we shall consider the progression as an increasing one, since all the results which we obtain can be immediately applied to a decreasing series by changing the sign of d. 179. To find the sum of n terms of a series in arithmetical progression. Write the same series in a reverse order, and we have S= 1 + (1 - d) + (1 — 2d) + Adding, 2 S = (a+1) + (a + 1) + (a + 1) +- + a = n(a) since the series consists of n terms. Hence, if any three of the five quantities a, l, d, n, S, be given, the remaining two may be found by eliminating between equations (1) and (2). It is manifest from the above process that The sum of any two terms which are equally distant from the extreme terms is equal to the sum of the extreme terms, and if the number of terms in the series be uneven, the middle term will be equal to one-half the sum of the extreme terms, or of any two terms equally distant from the extreme terms. Example 1. Required the sum of 60 terms of an arithmetical series, whose first term is 5 and common difference 10. Here, a=5, d=10, n=60 .. l=a+(n-1)8=5+59 × 10=595 2 =600×30=18000= sum required. Example 2. A body descends in vacuo through a space of 16 feet during the first second of its fall, but in each succeeding second 324 feet more than in the one immediately preceding. If a body fall during the space of 20 seconds, how many feet will it fall in the last second, and how many in the whole time? To insert m arithmetical means between a and b. Here we are required to form an arithmetical series of which the first and last terms, a and b, are given, and the number of terms =m+ 2; in order then to determine the series we must find the common difference. Eliminating S by equations (1) and (2), we have Ex. 4. Required the sum of the odd numbers 1, 3, 5, 7, 9, &c. continued to 101 terms? Ans. 10201. Ex. 5. How many strokes do the clocks of Venice, which go on to 24 o'clock, strike in the compass of a day? Ans. 300. Ex. 6. The first term of a decreasing arithmetical series is 10, the common difference, and the number of terms 21; required the sum of the series? Ans. 140. Ex. 7. One hundred stones being placed on the ground, in a straight line, at the distance of 2 yards from each other; how far will a person travel, who shall bring them one by one to a basket, which is placed 2 yards from the first stone? Ans. 11 miles and 840 yards. GEOMETRICAL PROGRESSION. 180. A series of quantities, in which each is derived from that which immediately precedes it, by multiplication by a constant quantity, is called a Geometrical Progression. Thus, the numbers 2, 4, 8, 16, 32, in which each is derived from the preceding by multiplying it by 2, form what is called an increasing geometrical progression; and the numbers 243, 81, 27, 9, 3, . . . in which each is derived from the preceding by multiplying it by the number decreasing geometrical progression. 1 3' form what is called a The common multiplier in a geometrical progression, is called the common ratio. Generally, if a be the first term, and ẹ the common ratio, the successive terms of the series will be of the form, The exponent of ୧ in the second term is 1, in the third term is 2, in the fourth term 3, and so on; hence, the nth term of the series will be of the forin, 181. To find the sum of n terms of a series in Geometrical Progression. If the series be a decreasing one, and consequently g fractional, it will be convenient to change the signs of both numerator and denominator in the above expressions, which then become, ୧ Hence, it appears, that if any three of the five quantities, a, l, g, n, S, be given, the remaining two may be found by eliminating between equations (1) and (2). It must be remarked, however, that when it is required to find e from a, n, S given, or from n, 1, S given, we shall obtain g in an equation of the nth degree, which cannot be solved generally. Example 1. Required the sum of 10 terms of the series 1, 2, 4, 8, Here, a = 1, g = 2, n = 10 a (en - 1) ... |