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-1. Write down a few of the leading terms of the scries, and place their indices over them, boginning the indices with an unit or 1.4 2. Add together such indices, whose sum shall make the entire index to the sum required.

3. Multiply the terms of the geometrical series belonging to those indices together, and the product will be the teria sought.

EXAMPLES. 1. If the first be 2, and the ratio 2; what is the 13th teum. 1, 2, 3, 4, 5, indices. Then 5+5+5=131 2, 4, 8, 16, 32, leading terms. 32X32X8=8192 Ans.

2. A draper sold 20 yards of superfine cloth, the first yard for Sd. the second for Oil, the third for 21d. &c. in triple proportion geoinetrical ; what did the cloth come. to at that rate ?

T!ie 20th, or last term is $486784401d. Then 3-4-5-186784401--3:

5230176600d. the sum of all

3-1 the terms (by Prob. I.) equal to £21792402 10s. Ans.

3. A rich miser thought 20 guineas a price too much for 12 fine horses, but agreed to give 4 cents for the first, 16 cents for the second, and 64 cents for the third horse, and so on in quadruple or fourfold proportion to the lastri what did they come to at that rate, and how much did they cost per head, one with another, ?

Ans. The 12 horses came to 8225696, 20cts, and the average price was $18641, 55cts. per head.

product of any two terms is equal to that term, signified by the sum of their indices. Thus,

$ 1 2 3 4 5 fc. Indices or arithmetical series

224 8.16 32 &c. geometricai series. Now, 5+2 5 = the index of the fifth term, and

4 X8 = 52 - the fifth term

CASE II. When the first terin of the series and the ratio are diffe

rent, that is, when the first term is either greater or less than the ratio.*

1. Write down a few of the leading terins of the series, and begin the indiees with a cypher: Thus, 0, 1, 2, 3, &c.

2. Add together the most convenient indices to make an index less by 1 than the nuinber expressing the place of the term svught.

3. Multiply the terms of the geometrical series togetirer belonging to those indices, and make the product a dividend.

4. Raise the first term to a power whose index is one less than the number of the terms anultiplied, and make the result a divisor.

5. Divide, and the quotient is the term sought:


4. If the first of a geometrical series be 4, and the ratio 3, what is the 7th term ?

0, 1, 2, S, Indices.
4, 12, 50, 108, leasing terns.

3+9+1=6, the index of the i th term.

=2916 the 7 th term required.

*16 Here the number of terms multiplied are three; there. fore the first term raised to a power less than three, is the 21 power or square of 4=16 the divisor.

*When the first term of the series and the ratio are different, the indices must begin with a cypher, and the surr of the indices made choice of must be one less than the numbar of terus given in the question : because 1 in the indices stands over the second term, and 2 in the indices over the thirA term, fc. and in this case, the product of any two terms, dirided by the first, is equal to that term beyond the first, signified by the sum of their indices. Thus,

So, 1, 2, 3, 4, &c. Indices.

21, 3, 9, 27, 81, &c. Geometrical series. jlrre 4.+3=7 the index of the Sth term.

SIX=2187 the 8th term, or the 7th beyond the 1st.


5. A Gullsmithi sold 1 lb. of gold, at 2 cents for the first once, 8 cents for the second, 32 cents for the third, &c. in quadruple proportion geometrically: what did the whole come to ?

Jus. S111848, 10cts. 3. Wat 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 old proportion ?

Ans. £113740740 14s. 9d. Sqr's. 7. A thresher worked 20 days for a farmer, and received for the first day's work four barley-corns, for the second 12 bailey-coros, for the third 36 barley-corns, and so on in triple proportion geometrical. I demand what the 20 olays' labor came to, supposing a pint of barley to contain 7680 .corns, and the whole quantity to be sold at 2s. 6il. per bushel. Ans. £1773 is. Eel. rejecting remainders.

8. A man bought a horse, and by agreement was to give a tarthing for the first nail, two for the second, four for the thical, ác. There were four shoes, and eight nails in each shoe; what did the lorse come to at that rate ?

Ans. 64473924 5s. Sjd. 9. Suppose a certain body, put in motion, should more 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. ift. lin. 16.c. which is no less than fire hundred and forty-one millions of miles.

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

Is when one nuinber is required, the properties of
who are given in the question.


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.


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 ? Suppose he had 12 As 37 : 148 : : 12 : 48 Ans as many 12

as inany

as many

as many

Result, S7

Proof, 148 2. What number is that which lcing increased by 5, 1, and' 1 of itself, the sum will be 125 ? Ans. 60.

3. Divide 93 dollars between A, B and C, so that B’s share may be lialf as much as A's, and C's sliare three times as much as B's."

Ans. It's share 31, B's 15), and C's 464 dolls. 4. A, B and C, joined their stock and gained 360 dols. of which A took up'a certain sum, B took 3 times as sbuch 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 7311. principal and interest together : What was the sumi delivered to lim at first?

Ans. £425. 6. A vessel has 3 cocks, A, B and C; A can fill it in 1 hour, B in 2 hours, and C in 4 hours ; in what time will they all fill it together?

Ans. 34 min. 17 sec.


DOUBLE POSITION, TEACHES to resolve questions by making two suppositions 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 froin 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 differenco of the errors, and the quotient will be the answer.

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

Nore. 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 sinall.


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 niore 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 À 8 B 10

B 12 C 18

(20) D S6

D 40

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* Those jitestions, in which the results are not propor. tional to their positions, belong to this rule ; such as thusa, in which the number sought is increased or diminished by some giren number, which is no known part af the number required

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