4. A body falls down an inclined plane 45 feet in 3 seconds. How many feet would it fall in 7 seconds? Distance

(time)2.

5. Two parallelograms have a common angle, and their adjacent sides are respectively 14 and 15, and 16 and 28. The area of the former is 96; find that of the latter.

6. A body weighs 180 pounds on the earth's surface. How many pounds would it weigh 500 miles above the earth's surface? Weight is inversely proportional to the square of the distance from the center of the earth. Radius 4000 miles.

7. Neptune is 30.05 times as far from the sun as the earth. Find its period of revolution. (Period)2 ∞ (distance). Earth's 365 da.

=

8. A pendulum vibrates 5 times in 4 seconds; find its length, a second's pendulum being 39.37 inches and Τα length.

9. The earth's diameter is 7920 miles, and the moon's, 2160 miles; the earth's mass is 80 times that of the moon, and the value of g on the earth's surface is 32.2 feet. Find g on the moon's surface. go

mass

(radius)2

189. The points of an unlimited straight line may represent all real numbers. To show this, take an un

limited straight line X'X and

mark on it a point 0, then

take any convenient unit of length and measure off on the line parts equal to this unit, marking them as indicated.

X'

‚ -8, -7, -6, -5, -4, -3, -2, -1, 0 1, 2, 3, 4, 5, 6, 7, 8,.....

X

Positive and negative fractions occupy intermediate positions. Thus 2 lies between 2 and 3. Every point on X'X represents some number, and every number is represented by a point.

A number a is greater than a number b (a >b) if a follows b in this scale. A number a is less than a number b (a < b) if a precedes b in the number scale.

From this statement it readily follows that if :
1. a>b>c, then a> c.

2. a<b<c, then a < c.
3. ab, then a±n>b±n.

4. ab, then a±n<b±n.

5. ab and c>d, then a+c>b+d.
6. ab and c<d, then a + c <b+d.
7. ab and c>0, then ac>bc.

8. ab and c>0, then ac<bc.

On the other hand, if c is a negative number, i.e. if 9. ab and c<0, then acab.

10. ab and c<0, then ac> ab.

Multiplication by a negative number reverses the sign of inequality, or, as it is sometimes said, it changes the sense of the inequality. The statement of these theorems in words, an important matter, is left as an exercise for the student.

190. If a, b, and x are positive numbers, then

The second part is proved in a similar manner.

191. If a, b, c, etc., are positive numbers, and if

a+c+ e

lies be

b+d+f

are not all equal to one another, then
tween the greatest and the least of the fractions.

I.e.

Adding 19,

(3, § 189)

Subtracting 5x, 26 > 2 x. .. 13 > x, or x < 13.
The inequality holds for values of x less than 13.
(b) Solve 2x2+ 3x < 27.

2x2+3x-27 <0, or (x-3) (2x+9) < 0.

This inequality holds for values of x that make x — 3 negative and 2x+9 positive... - 41 <x<3.

(c) If a and b are two numbers, real and unequal, then a2 + b2>2 ab.

For (a - b)2 > 0, i.e. a2 — 2 ab + b2 > 0.

... a2+b2>2 ab.

(d) If a, b, c are real and unequal, then

a2 + b2 + c2 > ab + bc + ca.

(3, § 189)

For a2 + b2 > 2 ab, b2 + c2 > 2 bc, c2 + a2 > 2 ca. Adding and dividing by 2, a2 + b2 + c2 > ab + bc + ca.

EXERCISE 145

1. If a b and n is a positive number, compare :

2. If

y and x > 0, y > 0, show that x2 < y2.

3. If xy and x < 0, y < 0, show that x2>y2.

a

b

n

4. If a > 0, b>0, show that + > 2, unless a = b.

b

5. If a > 0, b > 0, show that (a + b) > √ab >

6. Solve 3x-9 < x+2; x + 11 < 5x-53;

7. For what values of n is n > n2 - 7 n + 12?
8. For what values of n is n + 11 > n2 — 9 n?

9. If a, b, c are positive real numbers, prove

2 ab

a+b'

a3 + b3 + c3 > 3 abc. (See Ex. 25, Exercise 123.) 10. If a2+b2 1, x2+ y2 = 1, prove that ax + by < 1.

=

11. If > prove b d'

CHAPTER XVIII

ARITHMETICAL AND GEOMETRICAL PROGRESSIONS

193. A set of real quantities in which the nth quantity is uniquely determined if n is known is called a series. Illustrations:

a, a +d, a + 2 d, a + 3 d, a + 4 d, etc.,

a, ar, ar2, ar3, art, etc.,

1, 1, 1, 1, 1, 1, etc.

194. An Arithmetical Progression (A. P.) is a series in which each term differs from the one preceding by a constant quantity. This constant quantity is called the com

mon difference.

Illustrations:

1, 3, 5, 7, 9,

9, 5, 1,

3,

...

common difference 2.
7, ... common difference -4.

195. Fundamental formulæ. Denote the common difference by d, the first term by a, the number of terms by n, the nth or last term by l, and the sum of the series by 8. The general type of an A. P. is

a, a +d, a +2d, a + 3d, a + 4 d,....

/= a + (n-1) d.

Hence, Also 8 = a + (a + d) + (a + 2 d) + ... (1 − d) + 1, and 8 = l + (1 − d) + (1 − 2 d) + ··· (a + d) + a. .. 28 = (a + 1) + ( a + 1) + ( a + 1) ... to n terms

= n(a + 1).

...

(I)