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place of the term sought. Multiply the terms of the geometrical series together belonging to these indices, and make the product a dividend. Raise the first term to a power, the index of which is one less than the number of terms multiplied, and make the result a divisor, by which divide the dividend, and the quotient will be that term, beyond the first, signified by the sum of those indices, or the term sought.

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1. If the first term be 2, and the ratio 2, what is the 13th term?

1, 2, 3, 4, 5+5+3=13

2, 4, 8, 16, 32x32X 8-8192, Ans.

2. A merchant wishing to purchase a cargo of horses for the West Indies, a jockey told him he would take all the trouble and expense upon himself, of collecting and purchasing 30 horses for the voyage, if he would give him what the last horse would come to, selling them at the rate of two farthings for the first, four for the second, eight for the third, &c. What was the cost of the last horse? and what was the average price of the horses?

Ans. £1118481 1 s. 4 d., price of the last horse. £37282 14 s. 04 d., average price of the horses.

3. If the first is 5, and the ratio 3, what is the 7th term? Ans. 3645.

4. A man bought 20 cows, paying 2 farthings for the first, 10 for the second, and so on, in a five-fold ratio. What was the price of the last cow?

Ans. £39736429850 5 s. 2 d.

THE FIRST TERM, THE RATIO, AND NUMBER OF TERMS GIVEN, TO FIND THE SUM OF THE SERIES.

RULE.-Raise the ratio to a power, the index of which shall be equal to the number of terms, from which subtract 1; divide the remainder by the ratio less 1, and the quotient, multiplied by the first term, will give the sum of the series.

1. If the first term is 5, the ratio 3, and the number of terms 7, what is the sum of the series?

Divide by

less 1,

Ratio, 37-2187=7th power of the ratio. Subtract 1

the ratio 3-1=2|2186

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2. What debt can be discharged in a year, by paying 1 cent the first month, 10 cents the second, and so on, increasing in a ten-fold proportion each month?

Ans. $1111111111,11.

3. What is the sum of the infinite series, 1, 64, &c.?

Ans..

4. What is the sum of the infinite series,1, 01, 001, &c.? Ans.

THE EXTREMES AND THE RATIO GIVEN, TO FIND THE SUM OF THE SERIES.

RULE.-Divide the difference of the extremes by the ratio less 1, add the greater extreme to the quotient, and the result will be the sum of all the terms.

1. A gentleman, whose daughter was married on a newyear's day, gave her a guinea, promising to triple it on the first day of each month in the year; to what did her portion amount? Ans. 265720.

2. Suppose a ball to be put in motion by a force which impels it 10 rods the first minute, 8 the second, and so on, decreasing by a ratio of 1,25 each minute to infinity; what space would it move through? Ans. 50 rods. 3. What is the value of ,999, &c. to infinity? Ans. 1.

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Position is a method of performing such questions as cannot be resolved by the common direct rules, and is of two kinds, called single and double.

SINGLE POSITION.

SINGLE POSITION TEACHES TO RESOLVE THOSE QUESTIONS WHOSE RESULTS ARE PROPORTIONAL TO THEIR SUPPOSITIONS.

RULE.-Take any number, and perform the same operations with it, as are described to be performed in the question. Then say, as the result of the operation is to the position, so is the result in the question to the number required.

1. A's age is double that of B, and B's is triple that of C, and the sum of all their ages is 140; what is each person's age?

Suppose A's age to be 60
Then will B's: 230
And C's

3,0 =

10

100

As 100 60: 140:=84=A's age.
Consequently 84-42-B's "

And 42=14=C's "

140

2. A certain sum of money is to be divided between four persons in such a manner, that the first shall have of it, the second, the third, and the fourth the remainder, which is £28; what is the sum?

Ans. £112.

3. A person, after spending and of his money, had $60 left; what sum had he at first?

Ans. $144.

Ans. 60.

4. What number is that which, being increased by,, of itself, the sum shall be 125?

and

5. A person bought a chaise, horse and harness for £60; 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. £13 6s. 8 d. for the horse, £6 13 s. 4 d. for the harness, and £40 for the chaise.

6. A vessel has three cocks, A, B, and C; A can fill it in 1 hour, B in 2, and C in 3; in what time will they all fill it together?

Ans.

hour.

DOUBLE POSITION.

DOUBLE POSITION TEACHES TO RESOLVE QUESTIONS BY MAKING TWO SUPPOSITIONS OF FALSE NUMBERS.

RULE.-Take any two convenient numbers, and proceed with each according to the conditions of the question. Find how much the results are different from the result in the question. Multiply each of the errors by the contrary supposition, and find the sum or difference of the product. If the errors be alike, divide the difference of the products by the difference of the errors, and the quotient will be the answer. If the errors be 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 little; and unlike, when one is too great, and the other too little.

1. A lady bought tabby at 4 s. a yard, and Persian at 2 s. a yard; the whole number of yards she bought was eight, and the whole price 20 s.; how many yards had she of each sort?

Suppose 4 yards of tabby, value 16 s.
Then she must have 4 yards of Persian, value 8

Sum of their values 24

So that the first error is + 4

Again, suppose she had 3 yards of tabby at 12 s.
Then she must have 5 yards of Persian at 10

Sum of their values 22

So that the second error is + 2

Then, 4-2-2-difference of the errors.

Also, 4x2-8=product of the first supposition and second error.

And 3×4-12-product of the second supposition by the first error.

And 12-8-4-their difference.

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Whence 4-2-2-yards of tabby, Ans.
And 8-2-6 yards of Persian, S

2. Two persons, A and B, have both the same income; A saves of his yearly; but B, by spending $50 a year more than A, at the end of four years finds himself $100 in debt; what is their income, and what do they spend a year? Ans. Their income is $125; A spends $100, and B $150. 3. Two persons, A and B, lay out equal sums of money in trade; A gains $126, and B loses $87, and A's money is now double that of B; what did each lay out? Ans. $300.

4. A laborer was hired for 40 days on this condition, that he should receive 20 d. for every day he wrought, and forfeit 10 d. for every day he was idle: now he received at last £2 1 s. 8 d. How many days did he work, and how many was he idle? Ans. He wrought 30 days, and was idle 10.

5. A gentleman has two horses of considerable value, and a saddle worth £50; now, if the saddle be put on the back of the first horse, it will make his value double that of the second; but if it be put on the back of the second, it will make his value triple that of the first; what is the value of each horse? Ans. One £30, and the other £40.

6. There is a fish whose head is 9 inches long, and his tail is as long as his head and half as long as his body, and his body is as long as his tail and his head; what is the whole length of the fish? Ans. 6 feet.

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