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GEOMETRICAL PROGRESSION, Is when any rank or serise of numbers increased by one common multiplier, or decreasea by one common divisor; as 1, 2, 4, 8, 16, &c. in rease by the multiplier 2; and 27, 9, 3, 1, decrease by the divisor 3.

PROBLEM I.

The first term, the last term (or the extremes) and the ratio given, to fini! 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 3, what is its sum total ?

3x 1458-2

=2186 the Answer.

3-1 2. The extremes of a geometrical series are 1 and 65536, and the ratio 4 ; what is the suin of the series ?

Ans. 87381 PROBLEM JI. Given the first term, and the ratio, to find any other term

assigned.*

CASE 1. When the first term of the series and the ratio are equal.t

series of

* As the last terin 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 numbers in arithmeticat proportion called indices, whose common difference is 1.

+ When the first terin of the sortus mil the ratio are equal, che indices must begin with the init, und in this case,

the

1. Write down a few of the leading terms of the series, and place their indices over them, beginning the indices with an unit or 1.

*. Add together such indices, whose sum shall make up 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 term sought.

EXAMPLES.

1. If the first be 2, and the ratio 2 ; what is the 13th term. 1, 2, 3, 4, 5, indices.

Then 5+5+3=13 2, 4, 8, 16, 32, leading terms. 32X32 - 83-8192 Ans.

2. A draper sold 20 yards of superfine cloth, the first yard for 3d. the second for 9d. the third for 27d. &c. in triple proportion geometrical ;, what did the cirth come to at that rate :

The 20th, or last term is 3486784401d. Then 3+3486784401-3

5230176600d. the sum of all

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

3. A rich miser thouy ht 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 80 on in quadruple or tourfold proportion to the last : 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 $223696, 20cts. and the average price was $18641, 35cts. per head.

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

» 1 2 3 4 5 &c. Indices or arithmetical series.

22 4 8 15 32 fr. geometrical series Nowo,

3+2= 5.- the index of the fifth term, and 4X8 33 the fifth term.

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CASE II. When the first term of the series and the ratio are differ

ent, that is, when the first term is Rither greater or less than the ratio. *

1. Write down a few of the lending terms of the series, and begin the indices 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 number expressing the place of the term sought. 3. Multiply the terms of the geometrical series together

indices, and make the product a dividend. 4. Rawe the irst term to a power whose index is one less than the number of the terms multiplied, and make the esult a divisor. 5. Divide and the quotient is the term sought.

EXAMPLES. 4. If the first of a geometrical series be 4, and the ra. jo 2, what is the 7th term ?

0, 1, 2, 3, Indices.
4, 12, 36, 108, leading terms.

3+2+1=6, the index of the 7th term. 108 X36X12-46656

-=2916 the 7th term required.

16 Here the number of terms multiplied are three; therefore the first term raised to a power less than three, is the 2d 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 sum of the indices made choice of must be one less than the number of terms given in the question : because 1 in the indices stands over the second term, and 2 in the indices over the third term, fc. and in this case, the product of any two terins, divided by the first, is equal to that term beyond the first, signified by the sun of their indices. Thus,

> 0, 1, 2, 3, 4, &c. Indices.

1, 3, 9, 27, 81, $c. Gcometrioai series. Here 4+3.=7 the index of the 8th tcrm.

81X272187 the 8th term, or the 711 beyond the 1t.

SO on

5. A Goldsmith sold i lb. of gold, at 2 cts for the first ounce, 8 cents for the second, 32 cents for the third, &c. in a quadruple proportion geométrically: what did the whole come to?

Ans $111848. 10cts. 6. What debt can be scharged in a year, by paying 1 farthing the first month,

Things, or (2 d.) the second, and so on, each month'in a tento proportion ?

Ans. £115,1074 148 9d. 3qrs. 7. A thresher york: d 2 days former, and receive ed for the first day's work four harley currise for the second 12 barley-corns, for the third 36 barley-orn. in triple proportion geometrical. I demand whan 20 day's labour came to, supposing a pint of harley to conta 7680 Corns, and the whole quantity to be sold at 2s 60. per bushel Ans. £17737s 61. rejecting remainders.

87 A man bought a horse, and by agreement was to give a farthing for the first rail, two for the secor four for the third, &c. There were four shoes, and eigh nails in each shoe ; what did the horse come to at that rate ?

Ans. £4473924 5s. 3 d. 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 geumetrical ; how many yards would the said body move in che term of half a minute ?

Ans. 953199685623 yds. 1ft. lin. 16. which is no less than five hundred and forty-one millions of miles.

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POSITION POSITI

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

SINGLE POSITION, IS when one number is required, the properties of which are given in the question.

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

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

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

XAMPLES

1. A schoolmaster being asked how many scholars he had, said, If I had as many more as I now have, half as inany, 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 Anke as many 12

48
} as many
6

24
} as many
as many =

12

16

Result, 37

Proof, 148 2. What number is that which being increased by , and of itself, the sum will be 125 ?

Ans. 60. 3. "Divide 93 dollars between A, B and C, so that B's nare may be half as much as A's, and C's share three imes as much as B's.

Ins. A's share $31, B's $154, and C': $461, 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 much as A, and C touk 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 eceive interest for the same at 61 per centper annum, simple interest, and at the end of twelve years received 7311. principal and inter-st together; what was the sum delivered to him at first

An £425. 6. A vessel has 3 cock“, 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? Sins. 34 min. 171sec

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