ART. 343. To extract the cube root of a common fraction

Reduce the fraction to its simplest form, and extract the cube root of the numerator and denominator separately, if they are perfect cubes; if not, reduce the fraction to a decimal, and then extract the root.

Find the cube roots of the following:

1. A carpenter wishes to make a cubical cistern that wili contain 13824 cubic feet of water; what must be the length of each of its edges?

Ans. 24 feet. 2. A man has a pile of wood containing 864 cords. Suppose the wood to be piled in the shape of a regular cube, what would be the length of each edge? Ans. 48 feet.

3. Aallon of wine contains 231 cubic inches. What must be the depth of a cubical cistern to hold 2500 gallons of wine? Ans. 83.27 inches

A RITHMETICAL PROGRESSION.

ARTICLE 344. An ARITHMETICAL PROGRESSION is a series of numbers which increase or decrease by a common difference. When they increase they form an ASCENDING SERIES; when they decrease, a DESCENDING SERIES.

Thus, 2, 4, 6, 8, 16, 14, 12, 10, ART. 345. The numbers which form the series are called TERMS. The first and last terms are called the EXTREMES, and the other terms, the MEANS.

10, 12, etc., is an ASCENDING series; 8, 6, etc., is a DESCENDING series.

ART. 346. In Arithmetical Progression five particulars are to be considered: 1st, the first term; 2d, the last term; 3d the common difference; 4th, the number of terms; 5th, the sum of the series.

Any three of these quantities being known, the other two can be found.

CASE I.

ART. 347. The first term, the common difference and the number of terms being given, to find the LAST TERM. Analysis. Let us take the ascending series:

3, 5, 7, 9, 11, 13, 15, 17.

Here 3 is the first term, 17 the last term, 2 the common difference and 8 the number of terms.

In this series we see that the second term equals the first, plus once the common difference; the third term equals the first, plus twice the common difference; the fourth term equals the first, plus three times the common difference; the fifth term equals the first, plus four times the common difference; and, in general, any term equals the first term, plus the com. mon difference multiplied by the number of terms less 1 Hence, we have this

RULE.

Multiply the common difference by the number of terms less 1, and add the product to the first term.

NOTE. When the series is descending the product must be nubtracted from the first term.

the com.

Ex. 1. The first term of an ascending series is 5, mon difference 4 and the number of terms 10; what is the last term?

OPERATION.

5+ (101) × 45+9x45+36=41. Ans.

2. A boy bought 9 melons, paying 20 cents for the first, 24 cents for the second, 28 cents for the third, and so on; how much did he pay for the last? Ans. 52 cents.

3. A man discharged a debt by 7 different payments. The first payment was 25 dollars, and each succeeding payment was 10 dollars more than the preceding; what was the last payment? Ans. 85 dollars.

4. The first term of a descending series is 124, and the common difference 6; what is the 20th term? Ans. 10. 5. What is the 35th term of an ascending progression whose first term is 7 and common difference 5? Ans. 177.

CASE II.

ART. 348. The first term, the last term and number of terms being given, to find the COMMON difference.

From the last article it is evident that the difference between the two extremes is always equal to the common difference multiplied by the number of terms less 1; from which it follows that if we divide the difference between the extremes by the number of terms less 1, we shall have the commor difference. Hence, the

RULE.

Divide the difference between the extremes by the number of terms less 1.

Ex. 1. If the first term is 4, the last term 60 and the num ber of terms 9, what is the common difference?

2. If the last term is 76, the first term 6 and the number of terms 15, what is the common difference?

Ans. 5. 3. The extremes of a series are 5 and 89, and the number f terms is 13; what is the common difference? Ans. 7. 4. A man travels 13 days on a railroad, going 117 miles the first day, and 357 the last, increasing each day's travel by an equal difference; what is the daily increase? Ans. 20 miles.

CASE III.

ART. 349. The extremes and common difference being given, to find the NUMBER of terms.

Since the difference between the two extremes is equal to the common difference multiplied by the number of terms less 1, it follows that if we divide the difference between the two extremes by the common difference, the quotient will give the number of terms less 1; and if we add one to this quotient, the result will be the number of terms. Hence, we derive the

RULE.

Divide the difference between the two extremes by the common difference, and add 1 to the quotient.

Ex. 1. If the extremes of a series are 102 and 6, and the common difference 8, what is the number of terms?

OPERATION.

102- 6

96

+1

8

8

+1=12+1=13, number of term.

2. The extremes of a series are 118 and 10, and the common difference 12; what is the number of terms?

Ans. 10.

3. A man on a journey traveled the first day 15 miles, and

the last day 39 miles, each day increasing his distance 3 miles; how many days did he travel? Ans. 9 days.

4. The ages of the children in a certain family are in arithmetical progression, the common difference being 24 ycars. The oldest is 22, and the youngest 4 years of age; how many children in the family? Ans. 9 children.

CASE IV.

ART. 350. The extremes and number of terms being given, to find the SUM of the series.

Ex. 1. What is the sum of the series 2, 5, 8, 11, 14, 17?

Analysis. Let us take the series given in our example, and let us write under it the same series in an inverted order, and let us add the several terms in the same column:

2+ 5+ 8+11+14+17= 57 sum of first series. 17 +14 + 11+ 8+ 5+ 2 = 57: sum of inverted series. 19+19+19+19+19+19=114- =sum of both series.

We at once see that the sum of the terms in the two series is equal to 19 multiplied by 6 114. But 19 is the sum of the extremes, and 6 is the number of terms in each series.

= =

Since the sum of the extremes, multiplied by the number of terms, gives twice the series, it is obvious that if we divide this product by 2 we shall have the sum of the single series Hence the

RULE.

= 57.

Multiply half the sum of the extremes by the number of

terms.

2. The extremes of a series are 15 and 75, and the number of terms is 12; what is the sum of the scries?

Ans. 540.