EXAMPLES.

1. The extremes are 3 and 29, and t terms 14, what is the common difference?

29

-35

Number of terms less 1=13)26(2 Ans.

namber of

Extremes.

2. A man had 9 sons, whose se eral ages differed alike. the youngest was 3 years old, and the oldest 35, what was the common difference of their ages?

Ans. 4 years.

3. A man is to travel from New-London to a certain place in 9 days, and to go but 3 miles the first day, increasing every day by an equal excess, so that the last day's journey may be 45 miles: Required the daily increase, and the length of the whole journey?

Ans. The daily increase is 5, and the whole journey 207 miles.

4. A debt is to be discharged at 16 different payments (in arithmetical progression.) the first payment is to be 147. the last 100l.: What is the common difference, and the sum of the whole debt?

Ans. 5l. 14s. 8d. common difference, and 9121. the whole debt.

PROBLEM III.

Given the first term, last term, and common difference, to find the number of terms.

RULE.

Divide the difference of the extremes by the common difference, and the quotient increased by I is the number of terms.

EXAMPLES.

1. If the extremes be 3 and 45, and the common difference 2; what is the number of terms?

Ans. 22.

2. A man going a journey, travelled the first day five miles, the last day 45 miles, and each day increased his journey by 4 miles; how many days did he travel,

and how far?

Ans. 11 days, and the whole distance travelled 275 mites.

GEOMETRICAL PROGRESSION,

Is when any rank or series of numbers increased by one common multiplier, or decreased by one common divisor as 1, 2, 4, 8, 16, &c. increase by the multiplier 2; and 27, 9, 3, 1, decrease by the divisor S.

PROBLEM I.

The first term, the last term (or the extremes) and the ratio given, to find the sum of the series.

RULE.

Multiply the last term by the ratio, and from the product subtract the first term; then divide the remainder by the ratio, less by 1, and the quotient will be the sum of all the terms.

EXAMPLES.

1. If the series be 2, 6, 18, 54, 162, 486, 1458, and the ratio S, what is its sum total ?

2. The extremes of a geometrical series are 1 and 65536, and the ratio 4; what is the sum of the series? Ans. 87581.

PROBLEM II.

Given the first term, and the ratio, to find any other term

assigned.*
CASE I.

When the first term of the series and the ratio are equal.†

*As the last term in a long series of numbers is very tedious to be found by continual multiplications, it will be necessary for the readier finding it out, to have a series of numbers in arithmetical proportion, called indices, whose common difference is 1.

When the first term of the series and the ratio are equal, the indices must begin with the unit, and in this case, the

5. A Goldsmith sold 1 lb. of gold, at 2 cents for the first ounce, 8 cents for the second, 32 cents for the third, &c. in a quadruple proportion geometrically; what did the whole come to ? Ans. $111848, 10cts.

6. What debt can be discharged in a year, by paying 1 farthing the first month, 10 farthings, (or 24d.) the second, and so on, each month in a tenfold proportion ? Ans. 115740740 14s. 9d. Sqrs.

7. A thresher worked 20 days for a farmer, and received for the first day's work four barley-corns, for the second 12 barley-corns, for the third 36 barley-corns, and so on in triple proportion geometrical. I demand what the 20 days' labor came to, supposing a pint of barley to contain 7680 corns, and the whole quantity to be sold at 2s. 6d. per bushel? Ans. £1775 7s. 6d. rejecting remainders.

8. A man bought a horse, and by agreement was to give a farthing for the first nail, two for the second, four for the third, &c. There were four shoes, and eight nails in each shoe; what did the horse come to at that rate ? Ans. £4473924 5s. Sąd.

9. Suppose a certain body, put in motion, should move the length of one barley-corn the first second of time, one inch the second, and three inches the third second of time, and so continue to increase its motion in triple proportion geometrical; how many yards would the said body move in the term of half a minute?

Ans. 953199685623 yds. 1ft. lin. 1b.c. which is no less than five hundred and forty-one millions of miles.

POSITION.

POSITION is a rule which, by false or supposed nunf bers, taken at pleasure, discovers the true ones required It is divided into two parts, Single or Double.

SINGLE POSITION,

Is when one number is required, the properties of hich are given in the question.

RULE.

1. Take any number and perform the same operation with it, as is described to be performed in the question.

2. Then say; as the result of the operation is to the given sum in the question :: so is the supposed number: to the true one required.

The method of proof is by substituting the answer in the question.

EXAMPLES.

1. A schoolmaster being asked how many scholars he had, said, If I had as many more as I now have, half as many, one-third and one-fourth as many, I should then have 148: How many scholars had he ?

and

Suppose he had 12

as many = 12

6

= 4

as many

as many

as many

=

Result, 37

Proof, 148

Ans. 60.

2. What number is that which being increased by }, }, of itself, the sum will be 125 ? 3. Divide 95 dollars between A, B and C, so that B's share may be half as much as A's, and C's share three times as much as B's.

Ans. A's share 31, B's 15, and C's 46 dolls.

4. A, B and C, joined their stock and gained 360 dols. of which A took up a certain sum, B took 34 times as much as A, and C took up as much as A and B both; what share of the gain had each ?

Ans. A $40, B $140, and C $180. 5. Delivered to a banker a certain sum of money, to receive interest for the same at 6l. per cent. per annum, simple interest, and at the end of twelve years received 731. principal and interest together: What was the sum delivered to him at first? Ans. £.425.

6. A vessel has 3 cocks, A, B and C; A can fill it in 1 hour, Bin 2 hours, and C in 4 hours; in what time will they all fill it together? Ans, 34min. 17 sec.

DOUBLE POSITION,

TEACHES to resolve questions by making two suppo

sitions of false numbers.*

RULE.

1. Take any two convenient numbers, and proceed with each according to the conditions of the question. 2. Find how much the results are different from the results in the question.

3. Multiply the first position by the last error, and the last position by the first error.

4. If the errors are alike, divide the difference of the products by the difference of the errors, and the quotient will be the answer.

5. If the errors are unlike, divide the sum of the products by the sum of the errors, and the quotient will be the answer.

NOTE. The errors are said to be alike when they are both too great, or both too small and unlike, when one is too great, and the other too small.

EXAMPLES.

1. A purse of 100 dollars is to be divided among 4 men, A, B, C and D, so that B may have 4 dollars more than A, and C 8 dollars more than B, and D twice as many as C: what is each one's share of the money?

1st. Suppose A

6

2d. Suppose A 8

B 10

B 12

*Those questions, in which the results are not proportional to their positions, belong to this rule; such as those, in which the number sought is increased or diminished by some given number, which is no known part of the number required.

1