GEOMETRICAL PROGRESSION.

A GEOMETRICAL PROGRESSION is a ratio or series of numbers increasing by a constant multiplier, or decreasing by a constant divisor. Thus, 1, 2, 4, 8, 16, 32, &c. is an increasing geometrical ratio. And, 16, 8, 4, 2, 1, .5, &c. is a decreasing geometrical ratio. The ratio is the multiplier or divisor, by which the series is founded. In Geometrical Progression there are five denominations, any three of which being given, the other two may be found. 1st. The first term.

2d. The last term.

3d. The number of terms.

4th. The ratio.

5th. The sum of the series.

To raise a power or series of numbers by the ratio, we place the ratio at the left hand for the first power; this (first power) multiplied by the ratio (its square) gives the second power, the second by the ratio gives the third, and so on, until the power is 1 number less than the first term.

GENERAL RULE.

Raise the ratio to a power, one number less than the first term; and this power multiplied by the first term, the product will be the last term. If the first term be subtracted from the product of the last term and ratio, and the remainder divided by the ratio less 1, it will give the sum of the series. Or,

1. Raise the ratio to a power equal to the number of terms; 2. Subtract one from that power;

3. Multiply the remainder by the first term;

4. Divide this product by the ratio less one, the quotient will be the sum of the geometrical series.

Ex.-1. A man bought 12 yards of cloth, giving 4 cents for the first yard, 12 cents for the second, and so on, in a threefold ratio: what did he pay for the last yard, and what was the amount? Thus, 3×9X27 × 81 × 243 × 729

or, 729

243

2187

2916

1458

729

6561

1458

5103

2. What debt can be discharged in a year, by paying 1 cent the first month, 10 cents the second, and so on, each month in a tenfold proportion? $1111111111.11.

3. A man bought a horse, and by agreement was to give 1 cent for the first nail, 2 for the second, 4 for the third, &c.; there were 4 shoes, and 8 nails in each shoe: I demand what the horse was worth at that rate? Ans. $42949672.95. 4. A gentleman whose daughter was married on a new-year's day, gave her a dollar, promising to triple on the first day of each month in the year; to how much did her portion amount?

Ans. $265720. 5. A gentleman dying, left his estate to his five sons; to the youngest $1000, to the second $1500, and ordered, that each son should exceed the younger by the ratio of 1.5; what was the amount of the estate? Ans. $13187.50.

Questions. What is Geometrical Progression? The ratio is what? How many denominations does the rule embrace? What are their respective names? How do you raise the ratio to any given power? Recite the general rule.

UNITED STATES' DUTIES.

DUTIES are imposed by law on goods, wares, and merchandise, imported into the United States, at certain rates per centum ad valorem. When a duty is said to be ad valorem, it is meant that it is at a certain rate on the whole value of the goods. The term is used to distinguish this class of duties from those imposed on the quantity; as a duty upon the gallon, pound, barrel, cwt., ton, &c. A written account or catalogue of articles sent to a purchaser or factor, with the prices and charges annexed, is called an invoice.

The ad valorem rates of duty upon goods, wares, and merchandise, are estimated by adding 20 per cent. to the actual cost thereof, if imported from the Cape of Good Hope, or from any other place beyond it; and 10 per cent. if imported from any other place or country, including all charges; commissions, outside packages, and insurance excepted.

CASE 1.

To find the duty on any amount of goods, wares, or merchandise, at any given rate per centum ad valorem.

RULE.

Reduce the cost of the goods to Federal money, and add 20 per cent. if imported from or beyond the Cape of Good Hope, or 10 per cent. if imported from any other place. Then multiply the amount by the given rate per cent. and divide by 100, or remove the decimal point two figures to the left for the duty required.

EXAMPLES.

1. What will be the duty on an invoice of woollen goods,

imported from London, which cost £1250 10s. sterling, at

30 per cent. ad valorem?

£1250.5 sterling cost

4.44 £1 sterling

50020

50020

50020

10)5552.22,0 actual cost in federal money 555 22 2 ten per cent. added.

6107.442

30

Ans. $1832.23,260 duty required

2. What will be the duty on an invoice of silk goods, imported from France, which cost 2650 francs, at 20 per cent. ad valorem? Ans. $109.20,3+ 3. What will be the duty on an invoice of silk and cotton goods, imported from India, which cost 2500 rupees, at 25 per cent. ad valorem ? Ans. $375. 4. What will be the duty on an invoice of raisins, imported from Spain, which cost 640 piasters 4 reals 24 marvadies=51241 reals plate, at 40 per cent. ad valorem? Ans. $225.487.

CASE 2.

To find the duty on any amount of goods, wares, or merchandise, at any given rate per pound, gallon, &c.

RULE.

Multiply the given rate per pound, gallon, &c. by the number of pounds, gallons, &c. and the product will be the duty required.

EXAMPLES.

1. What will be the duty on 150 chests of Hyson tea, imported direct from China, in a vessel of the United States, weighing gross 11250lbs., tare 20lbs. per chest, at 40 cents per pound? Ans. $3300. 2. What will be the duty on 20 pipes of French brandy, fourth proof, containing 2520 gallons, at 48 cents per gallon? Ans. $1209.60. 3. What will be the duty on 25 hogsheads of brown sugar,

weighing 43750lbs. gross, allowing 12 per cent. tare, and 7lbs. per hogshead for draft or scalage, at .03 cents per lb.? Ans. $1150.38.

SINGLE POSITION.

SINGLE POSITION is the working of one supposed number, as if it were the true one, to find the true number.

RULE.

1. Take any number, and perform the same operations with it as are described in the question; then,

2. As the result of the operation

Is to the given number,

So is the assumed number
To the number required.

PROOF.

Perform on the number found, the operation described in the question, and the result will be the given number,

EXAMPLES.

1. Two men, A and B, having found a sum of money, disputed who should have it: A said the half, third, and one-fourth of the money made 130 dollars, and if B could tell how much was in it, he should have it all, otherwise he should have nothing-I demand how much was in the bag. Suppose 60 dollars As 65 130 :: 60

20 36 105

2. A, B, and C, talking of their ages, B said his age was once and half the age of A; and C said his age was twice and one-tenth the age of both, and that the sum of their ages was 93-what was the age of each?

Ans. A's 12, B's 18, and C's 63 years. 3. Seven-eighths of a certain number exceeds four-fifths by 6-what is that number?

Ans. 80.

4. A's age is double that of B, and B's is triple that of C's, and the sum of their ages is 140-what is the age of each? Ans. A's 84, B's 42, and C's 14 years.

5. A man carrying a purse of money in his hand, another asked him how much was in it; he answered he could not tell; but the third, fourth, and fifth of it made 94 dol7lars-how much was in the purse? Ans. $120.

·00

200180

6. A person lent his friend an unknown sum of money, to receive interest for the same at 6 per cent. per annum, simple interest, and at the end of twelve years received for principal and interest 860 dollars-what was the sum lent? *15-4800; 800 :: 860 Ans 500. Ans. $hyy.

7. A person bought a chaise, horse, and harness, for 100 dollars; the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness— what did he give for each?

Ans. $11.11 for the harness, $22.223 for the horse, and $66.66 for the chaise.

: 370: 100

DOUBLE POSITION is used for solving such questions as require two supposed numbers.

RULE.

Suppose two numbers, and work with each according to the conditions of the question.

Mark the errors with the signs plus and minus, that is to say, when the result proves that the supposed number exceeds the required one, mark the error plus (+); and when the result proves the supposed number to be less than the required one, mark the error minus (-): then place the less error under the larger, and the first supposed number opposite the error of the second operation, and the second supposed number opposite the error of the first operation, with the sign (×) between them; then multiply accordingly. If the errors be both plus, or both minus, divide the difference of the products arising from the errors being multiplied by the supposed numbers, by the difference of the errors: but if one be plus and the other minus, divide the sum of the said products by the sum of the errors; the quotient, in either case, will be the true nmmber or answer.