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ARITHMETICAL PROGRESSION.

When a series of numbers or quantities increase or decrease by a constant difference, it is called Arithmetical progression; as, 1, 2, 3, 4, 5 6; 1, 3, 5, 7, 9, 11; 6, 5, 4, 3, 2, 1; 11, 9, 7, 5, 3, 1. There are five things to be particularly attended to in Arithmetical Progression; the first term, the last term, the number of terms, the common difference, and the sum of all the terms.

CASE I.

The first term, common difference, and number of terms being given to find the last term, and sum of all the

terms.

RULE 1.-Multiply the number of terms, less one, by the common difference, and to that product, add the first term, the sum is the last term.

2. Add the first and last terms together, and multiply the sum by the number of terms, and half the product will be the sum of all the terms.

1. A person sold 40 yards of muslin at 2 cents for the first yard, 4 cents for the second, increasing 2 cents every yard, what did they amount to? Ans. $16.40.

OPERATION.

Nos. of terms 40 139 × 2 = 78 + 2 = 80 last term.

1st term or extreme

2

Last do. or second extreme 80

82 × 40 = 3280

$16.40

2. A butcher bought 75 sheep, and gave 6 cents for the first, 8 for the second, &c., what did he give for the last, and what did the whole number cost him?

Ans. For the last $1.54, the whole $60.

3. A travels uniformly at the rate of 6 miles an hour, and sets off upon his journey 3 hours and 20 minutes before B; B follows him at the rate of 5

hour, 6 the second, 7 the third, and so on. hours will B overtake A?

miles the first

In how many Ans. 8 hours.

CASE II.

When the first and last terms (or two extremes) are given to find the common difference.

RULE.-Divide the difference of the extremes by the number of terms less 1; the quotient will be the common difference.

1. If the ages of 12 persons are equally different, the youngest is 18 years and the eldest 40, what is the common difference of their age? Ans. 2 common difference.

Illustration of the above question.

40

18

12-1 =

11)22(2 common difference.
22

2. The extremes are 3 and 45, and the number of terms as 22, what is the common difference?

Ans. 2.

3. A man received "charity" from 10 different persons, the first 4 cents, the last 49 cents, what was the common difference, and what did the man receive?

Ans. he received $2.65; com. dif. 5 cts. 4. The extremes are 3 and 39, and the sum of the series 399; what is the common difference?

Ans. 2.

GEOMETRICAL PROGRESSION. Geometrical Progression is the increase of a series of numbers by a common multiplier, or decrease by a common divisor; as 2, 4, 8, 16, 32; 32, 16, 8, 4, 2; the ratio is the number by which the series increases or decreases.

CASE I.

To find the last term and sum of the series.

RULE.-Raise the ratio to the power whose index is 1 less than the number of terms given.

2. Multiply the product by the first term, and the result will be the last term.

3. Multiply the last term by the ratio; from the product subtract the first term, and divide the remainder by the ratio less 1, for the sum of the series.

1. If I buy 16 cords of wood, and agree to pay 2 cents for the first, 4 for the second, 8 for the third, &c., doubling the price to the last, what will it cost me?

1st. 2nd. 3rd. 4th.

power 1, 2, 3, 4, ratio 2, 4, 8, 16

fourth power 16

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2. A person at the birth of his son, deposited in bank 1 cent, towards his fortune, promising to double it at the return of every birthday, until he was 21 years of age, what was his portion? Ans. $20,971.51.

3. A gentleman consented to have his daughter married on New Year's day, and agreed to give her one dollar towards her portion, promising to double it, on the first day every month for one year, what was her portion?

of

Ans. $4095.

4. A thresher wrought 20 days and received for the first day's labour 4 grains of wheat; for the second 12; for the third 36, &c, how much did his wages amount to, allowing 7680 grains to make a pint, and the whole to be disposed of at $1 per bushel? Ans. $14187.

5. A sum of money is to be divided among 10 persons, the first is to have $10, the second $30 and so on in three fold proportion; what will the last have? Ans. $196830.

DUODECIMALS.

This rule is of great use to carpenters, joiners, &c. The mame is derived from the latin words duo, 12 and decem, 10, and as the ratio is 12, it may with propriety be termed Duodecimals. As the French and other European nations divide their inch into 12 equal lines, so our American artificers, suppose the inch to be divided as follows:

DENOMINATIONS OF DUODECIMALS.

12 fourths make 1 third, fourths marked thus 12 thirds make 1 second, thirds

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EXAMPLES IN ADDITION.

The ratio being 12 the rule is evident.

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MULTIPLICATION.-Observe the following rules.
Feet multiplied by feet produce feet.
Feet multiplied by inches produce inches.
Feet multiplied by seconds produce seconds.
Inches multiplied by inches produce seconds.
Inches multiplied by seconds produce thirds.
Seconds multiplied by seconds produce fourths.
1. Multiply 5 feet 6 inches by 2 feet 4 inches.

F I.

5.6

2.4

11.0

1.10.0

12.10.0

2. Multiply 8 feet 6 inches by 14 feet 9 inches.

CASE II.

Ans. 125 ft. 4 in. 6 s.

RULE.-Multiply by the component parts, as in compound multiplication, and take parts for the inches as in practice.

1. Multiply 208 feet 8 inches 4 seconds, by 24 feet 3 inches 9 seconds.

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2. Multiply 4 feet 7 inches by 6 feet 4 inches.

Ans. 29 ft. 0. in 4 s.

3. How many square feet are in a floor 48 ft. 6 in. long,

by 24 ft. 3 in. broad?

Ans. 1176 ft. 1 in. 6 s.

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