9. The extremes of a series are 1024 and 152741, and the ratio is 1: what is the sum of the series?

10. A merchant hired a clerk for a year, and agreed to pay him 1 mill the 1st month; 1 cent the 2d; 10 cents the 3d, and so on, increasing in a tenfold ratio for each successive month: what was the amount of his wages?

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11. What is the sum of the infinite series 1+1+1+1, &c.; that is, the descending series whose first term is 1 and the ratio 2? Ans. 2. 12. What is the sum of the infinite series 1++++++s4, &c. 612. To find the ratio, when the extremes and number of terms are given.

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Divide the greater extreme by the less, and extract that root of the quotient whose index is 1 less than the number of terms.

13. The extremes of a series are 3 and 192, and the number of terms 7: what is the ratio? Ans. 2.

14. What is the ratio of a series of 5 terms, whose extremes are 7 and 567?

Note.-Other formulas in arithmetical and geometrical progression might be added, but they involve principles with which the student is supposed as yet to be unacquainted. For a fuller discussion of the subject, see Thomson' Day's Algebra.

ANNUITIES.

613. The term annuity properly signifies a sum of money payable annually, for a certain length of time, or forever.

OBS. 1. Payments made semi-annually, quarterly, monthly, &c., are also called annuities. Annuities therefore embrace pensions, salaries, rents, &c. 2. When annuities remain unpaid after they are due, they are said to be forborne, or in arrears. The sum of the annuities in arrears, added to the interest due on each, is called the amount.

The present worth of an annuity is the sum, which being put at interest, will exactly pay the annuity.

3. Whẹn an annuity does not commence till a given time has elapsed, it is called an annuity in reversion; when it continues forever, a perpetuity.

4. In finding the amount of annuities in arrears, it is customary to reckon compound interest on each annuity from the time it is due to the time of payThe process therefore is the same as find'ng the sum of an ascending geometrical series. (Art. 611.) Hence,

ment.

614. To find the amount of an annuity in arrears.

Make the annuity the first term of a geometrical series, the amount of $1 for 1 year the ratio, and the. given number of years the number of terms; then find the sum of the series, and it will be the amount required. (Arts. 610, 611.)

GBS. When the payments are not yearly, for the amount of $1 for 1 year, use its amount for the time between the payments; and instead of the number of years, use the number of payments that have been omitted, and proceed as before. 1. What is the amount of an annuity of $100 which has not been paid for years, at 6 per cent. compound interest?

Solution.-100×(1.06)2=112.36; and (112.36×1.06)—100÷.06

$318.36, TABLE, showing the amount of annuity of $1, or £1, at 5, 6, and 7 per cent. for any number of years from 1 to 20.

Note.-Multiply the given annuity by the amt. of $1, for the given number of years found in the Table, and the product will be the amount required.

3. What will an annual rent of $75 amount to in 9 years, at 5 per cent.? 4. What is the amount of $200 forborne for 9 years, at 6 per cent.? 5. What is the amount of $350 forborne for 10 years, at 7 per cent.? 6. What is the amount of $1000 forborne for 20 years, at 6 per cent.?

615. To find the present worth of an annuity.

Find the amount of $1 annuity for the given time as before; then divide this amount by the amount of $1 at compound interest for the same time, multiply the quotient by the given annuity, and the product will be the present worth. If the annuity is a perpetuity, or to continue forever, multiply it by 100, divide the product by the given rate, and the quotient will be the present value required. OES. For the amount of $1 at compound interest, see Table, p. 271.

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7. What is the present worth of an annuity of $40 to continue 5-years, at per cent: compound interest? Ans. $173.178.

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8. What is the present worth of an annuity of $80 to continue forever, at per cent. ?

616. To find the present worth of an annuity in reversion.

Find the present worth of the annuity from the present time till its termination; also find its present worth for the time before it commences; the difference between these two results will be the prescnt worth required.

9. What is the present worth of $79.625 at 5 per cent., to commence in years and continue 6 years? Ans. $332.50.

PERMUTATIONS AND COMBINATIONS.

617. By Permutations is meant the changes which may be made in the arrangement of any given number of things.

The term combinations, denotes the taking of a less number of things out of a greater, without regard to their order or position.

618. To find how many permutations or changes may be made in the arrangement of any given number of things.

Multiply together all the terms of the natural series of numbers from 1 up to the given number, and the product will be the answer. 1. How many changes may be rung on 5 bells?

Ans. 120.

2. How many different ways may a class of 8 pupils be arranged? 3. How many different ways may a family of 9 children be seated? 4. How many ways may the letters in the word arithmetic, be arranged? 5. A club of 12 persons agreed to dine with a landlord as long as he could seat them differently at the table: how long did their engagement last?

619. To find how many combinations may be made out of any given number of different things by taking a given number of them at a time.

Take the series of numbers, beginning at the number of things given, and decreasing by 1 till the number of terms is equal to the number of things taken at a time; the product of all the terms will be the answer required.

6. How many different words can be formed of 9 letters, taking 3 at a time? Solution.-9X8X7=504. Ans. 504 words.

7. How many numbers can be expressed by the 9 digits, taking 5 at a time? 2. How many words of 6 letters each can be formed out of the 26 letters of the alphabet, on the supposition that consonants will form a word?

SECTION XIX.

APPLICATION OF ARITHMETIC TO GEOMETRY,

620. In the preceding sections abstract numbers have been applied to concrete substances, or to objects in general, considered arithmetically. On the same principle, geometrical magnitudes may be compared or measured by means of the numbers representing their dimensions. (Arts. 7, 516. Obs. 3.)

OBS. The measurement of magnitudes is commonly called mensuration.

MENSURATION OF SURFACES.

621. In the measurement of surfaces, it is customary to assume a square as the measuring unit, whose side is a linear unit of the same name. (Leg. IV. 4. Sch. Art. 257. Obs. 2.)

Note. For the demonstration of the following principles, see references. 622. To find the area of a parallelogram, also of a square. Multiply the length by the breadth. (Art. 285, Leg. IV. 5.) OBS. When the area and one side of a rectangle are given, the other side is found by dividing the area by the given side. (Art. 156.)

1. How many acres in a field 240 rods long, and 180 rods wide?

2. How many acres in a square field the length of whose side is 340 rods?

3. If the diagonal of a square is 100 rods, what is its area?

4. A rectangular farm of 320 acres, is a mile wide: what is its length? 623. To find the area of a rhombus. (Leg. I. Def. 18. IV. 5.) Multiply the length by the altitude or perpendicular height. 5. Find the area of a rhombus whose length is 20 ft., and its altitude 18 ft. 624. To find the area of a trapezium. (Leg. IV. 7.)

Multiply half the sum of the parallel sides by the altitude.

6. Find the area of a trapezium the lengths of whose parallel sides are 27 ft. and 31 ft., and whose altitude is 15 ft.

625. To find the area of a triangle. (Leg. IV. 6.)

Multiply the base by half the altitude or perpendicular height. 7. Find the area of a triangle whose base is 50 t., end its altitude 44 ft.

626. To find the area of a triangle, the three sides being given. From half the sum of the three sides subtract each side respectively; then multiply together half the sum and the three remainders, and extract the square root of the product.

9. What is the area of a triangle whose sides are 20, 30, and 40 ft. ? 10. How many acres in a triangle whose sides are each 40 rods?

627. To find the circumference of a circle from its diameter. Multiply the diameter by 3.14159. (Leg. V. 11. Sch.)

Note.-The circumference of a circle is a curve line, all the points of which are equally distant from a point within, called the centre. The diameter of a circle is a straight line which passes through the centre, and is terminated on both sides by the circumference. The radius or semi-diameter is a straight line drawn from the centre to the circumference.

11. What is the circumference of a circle, whose diameter is 20 ft. ? 12. What is the circumference of a circle, whose diameter is 45 rods

628. To find the diameter of a circle from its circumference. Divide the circumference by 3.14159.

OBS. The diameter of a circle may also be found by dividing the area by 7854, and extracting the square root of the quotient.

13. What is the diameter of a circle, whose circumference is 314.159 ft. ?

629. To find the area of a circle. (Leg. V. 11.)

; or, mut

Multiply half the circumference by half the diameter; or, tiply the square of the diameter by the decimal .7854.

15. What is the area of a circle, whose diameter is 50 rods? 16. Find the area of a circle 200 ft. in diameter, and 628.318 ft. in circum. 630. To find the side of the greatest square that can be inscribed in a circle of a given diameter.

Divide the square of the given diameter by 2, and extract the square root of the quotient. (Art. 581. Obs. 1.)

17. The diameter of a round table is 4 ft.; what is the side of the greatest square table which can be made from it?

631. To find the side of the greatest equilateral triangle that can be inscribed in a circle of a given diameter.

Multiply the given diameter by 1.73205. (Leg. V. 4. Sch.)

18. Required the side of an equilateral triangle inscribed in a circle of 20 ft, diameter,