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480y1-125y3+3923y2-7176y+25740=0,

contains the four values of y, and then x is found from the relation

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177. In discussing algebraical problems, it is frequently necessary to introduce inequations, that is, expressions connected by the sign 7. Generally speaking, the principles already detailed for the transformation of equations, are applicable to inequations also. There are, however, some important exceptions which it is necessary to notice, in order that the student may guard against falling into error in employing the sign of inequality. These exceptions will be readily understood by considering the different transformations in succession.*

I. If we add the same quantity to, or subtract it from, the two members of any inequation, the resulting inequation will always hold good, in the same sense as the original inequation. That is, if

a7b, then a+a' 7 b+a', and a—a' 7 b—a'•

* Example in Inequations.-The double of a number diminished by 6 is greater than 24; and triple the number diminished by 6 is less than double the number increased by 10. Required a number which will fulfil the conditions.

Let a represent a number fulfilling the conditions of the question; then, in the language of inequations, we have 2x-67 24, and 3x-6 / 2x+10. From the former of these inequations we have 2x 7 30, or x 7 15; and from the latter we get 3a—2x ≤ 10+6, or a 416; therefore 15 and 16 are the limits, and any number between these limits will satisfy the conditions of the question. Thus, if we take the number 15.9 we have 15.9 × 2-6 7 24 by 1.8, whilst 15·9 × 3—6 ▲ 15·9 × 2+10 by 1. Other examples may easily be formed

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The truth of this proposition is evident from what has been said with reference

to equations.

This principle enables us, as in equations, to transpose any term from one member of an inequation to the other, by changing its sign.

Thus, from the inequation,

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II. If we add together the corresponding members of two or more inequations which hold good in the same sense, the resulting inequation will always hold good in the same sense as the original individual inequations.

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III. But if we subtract the corresponding members of two or more inequations which hold good in the same sense, the resulting inequation WILL NOT ALWAYS hold good in the same sense as the original inequations.

Take the inequations 4≤7, 2≤3, we have still 4—2—7—3, or 2 ≤4. But take 9 10 and 68, the result is 9 — 6 (not <) 10 — 8, or 372.

We must therefore avoid as much as possible making use of a transformation of this nature, unless we can assure ourselves of the sense in which the resulting inequality will subsist.

IV. If we multiply or divide the two members of an inequation by a positive quantity, the resulting inequation will hold good in the same sense as the original inequation.

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This principle will enable us to clear an inequation of fractions.
Thus if we have

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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 877, multiplying both members by have the opposite inequation,

-24-21.

8

or

3

3'

Similarly, 87, but

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3, we

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

But,

ab then a">b".

Thus from 53 we have (5)2(3)2 or 259.

So also from (a+b)>c, we have (a+b)2>c2.

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.

Thus,

-243

2

gives (-2)2 (3) or 4 29

But, 3-5 gives (—3)*(— 5)2 or 9 25
Again, 3-5 gives (-3)3(-5) or -27 — 125.

In like manner,

IX. We can extract any root of both members of an inequation without altering the sense of the inequation. That is, if

ab, then, ab.

If the root be of an even degree both members of the inequation must neces sarily 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 d 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.

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Then S = a + (a + d) + (a + 2d) +--

Write the same series in a reverse order, and we have

d)

-----

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+ a

+(a + 1)

S= ? + (1-8) + (1—28) + Adding, 2 S = (a+1) + (a + 1) + (a + 1) + · = n(a+l) since the series consists of n terms.

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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.

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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?

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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

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