339. EXERCISES IN VARIATION.

1. If xy, and, when x = 5, y = 20, what is the value

7. If a b when c is constant, and a c

of a, b, and c in = 3 and c = 1?

8. If 4, 3, and 2 are simultaneous values Example 7, what is the value of a when b

9. If 8 men earn $400 in 5 weeks, find by Example 7 how many weeks it will take 6 men to earn $240 ?

10. The area of a triangle is measured by half the product of its base and altitude. Prove that in triangles of equal area the bases vary inversely as the altitudes.

11. If x3 ∞ y2, and x = 3 when y = 4, find the equation between x and y.

12. If xxy, and x = when y, what is x when

2, find

13. If a2 — b2 ∞ c2, and, when a = 5 and b = 3, c = the equation between a, b, and c, and prove that b is a mean proportional between a + 2c and a -2 c.

CHAPTER XXIII.

PROGRESSIONS.

340. A Progression is a series of numbers which increase or decrease according to some fixed law.

341. The Terms of a series are the successive numbers that form the series. The first and last terms are called the extremes, and the others the means.

ARITHMETIC PROGRESSION.

342. An Arithmetic Progression (A. P.) is a series in which each term, except the first, is obtained from the preceding by the addition of a constant number called the common difference. Thus, each of the following series is an arithmetic progression.

The first is an ascending progression, the second a descending progression.

The common differences of the series are, respectively, 3,-4,-2, and d.

The common difference of an arithmetic progression can be found by subtracting any term from that which immediately follows it.

343. In Arithmetic Progression there are five elements, any three of which being given, the other two can be

found.

1. The first term.

2. The last term.

3. The common difference.

4. The number of terms.

5. The sum of all the terms.

344. In Arithmetic Progression there are twenty possi In discussing this subject we shall let

ble cases.

345. The First Term, Common Difference, and Number of Terms given, to find the Last Term.

In this Case, a, d, and n are given, and 7 is required.

The successive terms of the series are

a,

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

that is, the coefficient of d in each term is one less than the number of that term, counting from the left; therefore, the last or nth term in the series is

in which the series is ascending or descending according as d is positive or negative. Hence,

Rule.

To the first term add the product formed by multiplying the common difference by the number of terms less one.

l= a + (n − 1) d = 2 + (171) 350 Ans.

:

346. The Extremes and the Number of Terms given, to find the Sum of the Series.

Now

In this Case, a, l, and n are given, and s is required.

or, reversing the series,

Adding these together,

28 =

· 1 + (l − a) + (1 − 2 d) + (l − 3 d) +

= (a + 1) + (a + 1) + (a + 1) + (a + 1) + ··· + (a + 1)

And since (a+1) is to be taken as many times as there are terms,

Find one half the product of the sum of the extremes and the number of terms.

NOTE. If in place of the last term the common difference is given, the last term must first be found by the Rule in Case I.

CASE III.

347. The Extremes and Number of Terms given, to find the Common Difference.

In this Case, a, l, and n are given, and

d is required.

Divide the last term minus the first term by the number of terms less one, and the quotient will be the common difference.

NOTE 1. This rule enables us to insert any number of arithmetic means between two given numbers; for the number of terms is two greater than the number of means. Hence, if m = the number of means, m + 2 = n, 1 -a or m1n — 1, and d = m+1

Having found the common differ

ence, the means are found by adding the common difference once, twice, &c., to the first term.

5. Find 6 arithmetic means between 4 and 39.

6. Find 4 arithmetic means between 17 and 52. 7. Find 6 arithmetic means between 2 and -26.

But ad is the second term of a series whose first term is a and common difference d, or the arithmetic mean of the series a, a + d, a + 2 d. Hence, the arithmetic mean between two numbers is one half of their sum.