REVIEW

WRITTEN EXERCISES

1. Solve, using the principle of composition and division,

(a−√2 bx + x2) : (a − b) = (a +√2 bx +x2) : (a + b).

3. Given x+√x: x−√x : : 3√x+6 : 2√ĩ, to find x.

4. If the volume of a sphere varies as the cube of its radius, find the radius of a sphere whose volume equals that of the sum of two spheres whose radii are, respectively, 6 ft. and 3.5 ft.

5. The number of vibrations (swings) made by two pendulums in the same time are to each other inversely as the square roots of their lengths. If a pendulum of length 39 in. makes 1 vibration per second (called a seconds pendulum), about how many vibrations will a pendulum 10 in. long make? How long must the pendulum be to make 10 vibrations per second?

6. Two towns join in building a bridge which both will use, and agree to share its cost, $5000, in direct proportion to their populations and in inverse proportion to their distances from the bridge. One town has a population of 5000 and is 2 mi. from the bridge; the other has a population of 9000 and is 6 mi. from the bridge. What must each pay?

3

y

what is the limiting value of x when y = ∞ ?

7. In x=

SUMMARY

The following questions summarize the definitions and processes treated in this chapter:

1. Define a proportion; define means; also extremes.

Secs. 198, 202.

2. What is a fourth proportional? A third proportional? A mean proportional? Secs. 199-201.

3. Define alternation; also inversion; composition; divi

Sec. 215.

7. Define and illustrate inverse variation.

8. State an equation expressing the law of direct variation;

also one expressing inverse variation.

a

∞

9. Define and illustrate limit.

Secs. 214, 216.

Sec. 219.

10. Explain the meaning of the symbol; also ∞; also

; also.

∞

Secs. 221-224.

11. What kind of a line is the graph of the equation expressing direct variation? Of the equation expressing inverse variation?

Sec. 225.

226. Series.

CHAPTER IX

SERIES

If a sequence of numbers is determined by a given law, the sequence of numbers is called a series.

227. Terms. The numbers constituting the series are called its terms, and are named from the left, 1st term, 2d term, etc. The following are examples of series:

1-12. State the next five terms of each series above.

228. When the law of a series is known, any term may be found directly.

EXAMPLES

1. The law of the second series in Sec. 227 is that each term is two To get the tenth term we start from 1 and

more than the preceding.

add 2 for each of 9 terms.

That is, the tenth term is 1 + 9.2

2. The law of the eighth series is that each term after the first is 3 times the preceding term. To get the ninth term we start at 1 and multiply by 3 eight times, or by 38. That is, the ninth term is 1.38 = 6561.

Similarly, the 12th term is 1. 311 the nth term is 1.3"-1

1-4.

WRITTEN EXERCISES

Select four of the series in Sec. 227 that can be treated like the first example and write the 10th, 12th, 15th, and 47th terms of each.

5-7. Select three of the series in Sec. 227 that can be treated like the second example and write the 8th, 10th, and nth terms of each.

8. Write similarly the 7th, 11th, 20th, 47th, and nth term for any 6 of the above series.

229. We shall give only two types of series, the arithmetical and the geometric, the laws of which are comparatively simple.

ARITHMETICAL SERIES

230. Arithmetical Series. A series in which each term after the first is formed by adding a fixed number to the preceding term is called an arithmetical series or arithmetical progression.

231. Common Difference. The fixed number is called the common difference, and may, of course, be negative.

For example:

1. 7, 15, 23, 31, 39,

...

ference 8.

is an arithmetical series having the common dif

...

2. 16, 14, 13, 111, 10, is an arithmetical series having the common difference.

WRITTEN EXERCISES

1. Select the arithmetical series in the list of Sec. 227. Write the nth term in each.

2. Beginning with 2 find the 100th even number.

3. Beginning with 1 find the 100th odd number.

4. Beginning with 3 find the 200th multiple of 3.

5. A city with a population of 15,000 increased 600 persons per year for 10 yr. What was the population at the end of 10 yr.?

232. A general form for an arithmetical series is:

a, a + d, a + 2 d, a + 3 d, ....., a + (n − 1)d, ......,

where a denotes the first term,

d denotes the common difference, and

n denotes the number of the term.

233. Last Term. If the last term considered is numbered n and denoted by 1, we have for the last of n terms the formula: l= a + (n-1)d.

234. The Sum of an Arithmetical Series. The sum of n terms of an arithmetical series can be found readily.

EXAMPLE

Find the sum of the first 6 even numbers.

1. Let

82 + 4 + 6 + 8 + 10 + 12.

2. We may also write s = 12+ 10 +8 + 6 + 4 + 2.

3. Adding (1) and (2),

2 s = (2 + 12) + (4 + 10) + (6 + 8) + (8 + 6) + (10 + 4) + (12 + 2) =6(2 + 12); for each parenthesis is the same as 2 + 12.

235. General Formula for the Sum. The general form of the series may be treated in the same way.