NOTE. --The term carat is a word used in indicating the proportion of pure gold in any given quantity of the metal; thus, if the metal be pure gold, it is said to be 24 carats fine; if two thirds gold, 16 carats fine; if 17 parts gold and 7 parts alloy, 17 carats fine, etc.

8. How much wool, of equal quantities, at 35 and 40 cents per lb., must be mixed with 100 lbs. at 60 cents per lb., that the mixture may be worth 45 cents per lb.?

484. When the entire quantity is limited, find the proportion of the ingredients as before, and then divide the given quantity among the ingredients in the proportion found.

EXAMPLES.

9. J. Blake has an order from New York for 1000 bushels of wheat, at $1.25 per bushel. How shall he mix his wheat, which he values at $1.20, $1.22, and $1.30, to fill the order?

Ans. 100 bu. at $1.20, 500 bu. at $1.22, 400 bu. at $1.30. 10. J. Smith wishes to purchase a farm of 200 acres, at $100 an acre. How much woodland at $125 per acre, mowing upland at $90 per acre, pasture land at $70 per acre, and tillage ground at $128 per acre, may he purchase?

11. How many lbs. of cotton at 60, 73, and 98 cents per lb., must be mixed with 750 lbs. at 90 cents, that the mixture may contain 2000 lbs. at 80 cents per lb.?

NOTE.First balance the loss on the 750 lbs. with gain on one of the other ingredients taken; then proceed to make a mixture of the other ingredients equal to the entire quantity given, minus the quantities bal, anced.

ARITHMETICAL PROGRESSION.

485. Arithmetical Progression is progression by equal differences.

486. An Arithmetical Series is a succession of numbers which increase or decrease by a common difference.

If the numbers increase from the first term, the series is an Increasing Series: e. g., 2, 4, 6, 8, 10, 12, &c.

If the numbers decrease from the first term, the series is a Decreasing Series; e. g., 13, 11, 9, 7, 5, &c.

487. In every series, five things are to be considered; viz., the First Term, the Last Term, the Number of Terms, the Common Difference, and the Sum of the Terms; any three of which being given, the other two may be found. This gives rise to twenty distinct cases, a few of the more important of which will be here presented.

NOTE I. For the remaining cases, also for full discussions of Geometrical Progression and Annuities, the student is referred to works on Algebra.

NOTE II. — Increasing series only will be considered in this book, as rules that apply to increasing series apply to decreasing series also, provided that, wherever the common difference is introduced, it is used with the contrary sign.

488. TO FIND ANY TERM IN A SERIES, WHEN THE FIRST TERM, COMMON DIFFERENCE, AND NUMBER OF TERMS

ARE GIVEN.

Let 5 terms.

first term, 2 common difference, and 6=the number of The series will be constructed as follows:

We find that the second term equals the first term, plus the common difference; the third term equals the first term, plus two times the common difference; the fourth term equals the first term, pius three times the common difference, &c.; and that the last or sixth term equals the first term, plus five times the common difference. Hence,

I. To find any term of the series: Add the first term to the product of the common difference multiplied by the number of terms which precede it.

II. To find the last term: Add the first term to the product of the common difference multiplied by the number of terms less one.

EXAMPLES.

1. In an increasing series the first term is 4, and the common difference is 8; what is the seventh term? Ans. 52.

2. The first term is 7, the common difference, and the number of terms 20; what is the last term?

3. If 5 lbs. of power is imparted to a fly-wheel at each revolution, what is its power at the end of the tenth revolution from a state of rest, provided its average loss of power from friction and other causes is 1 lb. during each revolution? Ans. 40 lbs.

4. If a stone, in falling to the earth, descends 16 feet during the first second, 3 X 16 feet during the next, 5 X 16 feet during the third, and so on; how far will it fall during the elev enth second?

5. What is the amount of $200 at simple interest for 8 years, cent.?

at 6

per

NOTE.-The amount will be the ninth term of the series, of which the first term is $200.

489. TO FIND THE COMMON DIFFERENCE IN A SERIES, ALSO THE NUMBER OF TERMS.

If, in series (1.) we subtract the first term from the last, we have res maining 5 × 2, that is, the common difference multiplied by the number of terms less one. Hence,

I. To find the common difference: Divide the difference between the first and last term by the number of terms less one.

II. To find the number of terms: Divide the difference between the first and last term by the common difference, and add one to the quotient.

EXAMPLES.

6. The first term of a series is 7, the last term 19. and the number of terms 13; what is the common difference? Ans. 1.

7. The first term is 30, the last term is 3, and the number of terms 10; what is the common difference?

8. The first term is 8, the last term 23, and the common dif ference 1; required the number of terms.

9 A boy, in picking up stones 2 feet apart, and carrying them, one at a time, to a deposit 2 feet from the first, found that to carry the last one, he had walked 60 feet; how many stones did in all? Ans. 15 stones.

he carry

490. To FIND THE SUM OF THE SERIES.

Let 2, 4, 6, 8, 10, 12, 14, 16, be a series, of which we wish to find We write under it the same series in an inverted order, and

the sum.
add the terms as follows:-

But 8 equals the number of terms, and 18 the sum of the extremes.

Hence,

To find the sum of a series: Multiply one half the sum of the extremes by the number of terms.

EXAMPLES.

10. The first term of a series is 4, the last 40, and the number of terms 11; what is the sum of the series? Ans. 242.

11. What is the sum of the odd numbers from 1 to 99 inclusive?

12. What is the sum of the multiples of 3 from 6 to 45 in. clusive?

13. How many notes must a person sing in ascending two octaves, if he goes back to the first note each time he strikes a new one, and sounds all the intermediate notes each time he ascends? Ans. 120 notes.

5

14. Two of Dio Lewis's pupils tried their skill in running for pegs. Each set up pegs 6 feet apart, and commenced running 6 feet from the first peg. How far did each run to place the pegs at his starting-point?

15. How far would the first boy of a row of 21 scholars travel, In gathering writing-books from the row, if the scholars were 21 feet apart, and he brought one book at a time to his own desk?

GEOMETRICAL PROGRESSION.

491. Geometrical Progression is progression by equal multipliers.

492. A Geometrical Series is a succession of numbers which increase or decrease by a common multiplier. Thus,

2, 4, 8, 16, 32, 64, is an increasing geometrical series, in which the multiplier is 2.

2, 1,,,,, is a decreasing geometrical series, in which the multiplier is.

493. The common multiplier is called the Ratio.

494. In every geometrical progression, five things are to be considered; viz., the First Term, the Last Term, the Number of Terms, the Common Ratio, and the Sum of the Terms; any three of which being given, the other two may be found.

495. TO FIND THE LAST TERM OF A SERIES, THE FIRST TERM, THE RATIO, AND NUMBER OF TERMS BEING GIVEN. Let 3 be the first term, 2 the ratio, and 5 the number of terms. The series will then become,

4th term.
3 X 23,

(1.) 1st term. 2d term. 3d term. 5th term. 3, 3 X 2, 3 X 22, 3 X 24, in which the second term equals the first term multiplied by the ratio, the third term equals the first term multiplied by the second power of the ratio, the fourth term equals the first term multiplied by the third power of the ratio, and the fifth term equals the first term multiplied by the fourth power of the ratio. Hence,

I. To find any term of the series: Multiply the first term by the ratio raised to a power equal to the number of terms which precede the required term.

II. To find the last term of the series: Multiply the first term by the ratio raised to a power equal to the number of terms less

one.