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GEOMETRICAL PROGRESSION,

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 froin 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, 169, 480, 1458, and the ratio 3, what is its sum totai?

SX1458---2

9156 thenseer.

S1 2. The extremes of a geometrical series are 1 and 65536, and the ratio 4; wliat is the sum of the series :

Ans. 8,981. PROBLEM II. Given the first term, and the ratio, to find any pther terin

assigned."

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

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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 rilimetical proportion, called indices, whose compierre eifference is 1.

{"!2z1 the first term of the seriestainid the ratio are equal, the in:lces must begin with the unit, si in this case, the

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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.

2. Add together such indices, whose sum shall make up the entire index to the sum required.

3. Multiply the terms of the geometrical series belong ing 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 13tia term. 1, 2, 3, 4, 5, indices, Then 51-54-3=15 2, 4, 8, 16, 32, leading terms. S2X32X8S192 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 cloth come to at that rate ?

The 20th, or last term is $486784401d. Then 3+3486784401-5.

==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 last what did they come to at that rate, and how much did they cest per head, one with another?

Ans. The 12 horses came to $223696, 20cts. and the average price eros $18641, 55cts, per head.

product of anytuo terms is equal to that term, signified by the sum of their indiees. Thus,

312 3 4 5 &c. Indices or arithmetical series

12 4 8 16 32 &c. geomeirical series. Now, 3+2 5 = the index of the fifth term, and

4x8 m 52 - the fifth tern.

CASE II. When the first term 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 terms of the series, and begin the indices with a cyplier: 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 belonging to those indices, and make the product a dividend.

4. Raise the first terın to a power whose index is one less than the number of the terms multiplied, and make the resulta divisor.

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

EXAMPLES.

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, 36, 108, leading terms.

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

==2916 the 7th term required.

16 Here the pumber of terms multiplied are three; there. fore the first term raised to a power less than three, is the 2d power or square of 4516 the shivisor.

When the first term of the series and the ratio are different, the indices must begin with a cypher, and the süin of the indices made choice of must be one less than the nunber of terms given in the queslim: because 1 in the indices stanis over the second terin, and 2 in the indices over the third tern, f'c. and in this case, the produce of any two terms, divided by the first, is equal to that term beyond the first, signified by the sain of their indices. 7'hus;

0, 1, 2, 3, 4, 5, Erediccs.

71, 3, 9, 27, 81, 8c. Geometrical series. Trip-S= the juda: of the 8th teru.

2137 the sih term, or the 7th bevorul tre. Est.

*

5. A Goldsmith sold 1 lb. of gold, at 2 cents for the first ounce, 8 cents for the second, 52 cents for the third, &c. in a quadruple proportion geometrically; what did the whole come to?

Ans. $111848, 10cts. S. What debt can be discharged in a year, by paying 1 farthing the first month, 10 farthings, (or 24d.) the seeond, and so on, each month in a old proportion ?

Ans. f.113740740 145. 9d. 3qrs. 7. A thresher worked 20 days for a farmer, and receive ed for the first day's work four barley-corns, for the second 12 barley-corns, for the third 56 barley-corns, and so on in triple proportion geometrical. I demani what the 20 days' labor came to, supposing a pint or barloy to contain 7689 corns, and the whole quantity to be sold at 25. 6d. per bushel ? Ans. £1775 7s. Gil. rejeciing remainders.

8. A man bought a horse, and by agreement was to give a farthing for the first vail

, two for the second, four for the third, &c. There were füur shoes, and eight nails in each shoe; what did the horse come to at that rate ?

Ans. £4475924 5s. Sid. 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 ?

Ins. 959190585623 yus. ift. lin. 16.c. which is no less than five hundred and forty-one nillions of iniles.

POSITION. 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.

SINGLE POSITION, Is when one number is required, the properties of

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.

13

1. A schoolmaster being asked how many scholars he had, said, If I had as many more as I now liave, half a many, one-third and one-fourth as many, I should then have 148: How many scholars had he? Suppose he had 19 As 37 : 148 : : 12 : 48 And as many

Aland

48
as many
6

24
} as many

16
# as many
S

19
Result, S7

Proof, 148 2. What number is that which being increased by 1, 1, and of itself, the sam will be 195? Ans. 60.

3. Divide 93 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.

3 share 31, B's 151, and C's 461 dolls. 4. A, B and joined their stock and gaineri 360 dols. of which a took up a certain suin, took 3} times as much as A, an!! C took up as much as A and what share of the gain had each :

as, o $40, B $140, and C $180. 5. Delivered to a banker a certain sum of money, to receive interest for the same at 01. per cent. per annum, simple interest, and at the end of twelve years received 731l. 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, B in 2 hours, and C in 4 hours; in what time will they all fill it together?

Ans. Sumin. 171 sec.

both;

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