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


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


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


1. If the series be 2, 6, 18, 54, 162, 486, 1458, the ratio 3, what is its sum total ?


=2186 the Answer


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

Ans. 8738


Given the first term, and the ratio, to find assigned.*


When the first term of the series and the ratio are equ


other t

*As the last term in a long series of numbers is ver dious to be found by continual multiplications, it wil necessary for the readier finding it out, to have a se of numbers in arithmetical proportion, called indi whose common difference is 1.

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

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 sam required.

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


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

Then 5+5+3=13
32x32x8=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 cloth come to at that rate?

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

1, 2, 3, 4, 5, indices.

2, 4, 8, 16, 32, leading terms.

⇒5230176600d. the sum of all the terms (by Prob. I.) equal to £21792402 10s. Ans.


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



1 2 3 4 5 &c. Indices or arithmetical series.
(2 4 8 16 32 &c. geometrical series.
3+2 5 = the index of the fifth term, and
4x8 = 32 the fifth term.


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When the first term of the series and the ratio are rent, that is, when the first term is either grea less than the ratio.*

1. Write down a few of the leading terms of the s and begin the indices with a cypher: Thus, 0, 1, 2,

2. Add together the most convenient indices to an index less by 1 than the number expressing the of the term sought.

3. Multiply the terms of the geometrical serie gether belonging to those indices, and make the pe a dividend.

4. Raise the first term to a power whose index is less than the number of the terms multiplied, and 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 S, 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.


2916 the 7th term required


Here the number of terms multiplied are three; the fore the first term raised to a power less than three, is 2d power or square of 4-16 the divisor.

*When the first term of the series and the ratio are
ferent, the indices must begin with a cypher, and the
of the indices made choice of wust be one less than the n
ber of terms given in the question: because 1 in the ind
stands over the second term, and 2 in the indices over
third term, &c. and in this case, the product of any
terms, divided by the first, is equal to that term beyand
first, signified by the sum of their indices.
(0, 1, 2, 3, 4, &c. Indices.

(1, 3, 9, 27, 81, &c. Geometrical series.
Here 4+3 7 the index of the 8th term.
81x272187 the 8th term, or the 7th beyond the 1

5. A Goldsmith sold 1 lb. of gold, at 2 cents for the first ounce, 3 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. £1773 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.

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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?

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


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


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

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1. Take any number and perform the same op with it, as is described to be performed in the quest 2. Then say; as the result of the operation is given sum in the question: so is the supposed nu to the true one required. The method of proof is by substituting the ans the question.


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

As 37 148: 12: 48


Suppose he had 12
as many 12
as many 1 6
as many → 4

as many





Result, 37

Proof, 148

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

Ans. E

3. Divide 93 dollars between A, B and C, so tha share may be half as much as A's, and C's share times as much as B's.

Ans. A's share $31, B's $15§, and C's 840 4. A, B and C, joined their stock and gained 360 of which A took up a certain sum, B took 3 time much as A, and C took up as much as A and B b what share of the gain had each ?

Ans. A $40, B $140, and C $18 5. Delivered to a banker a certain sum of mone receive interest for the same at 61. per cent. per anr simple interest, and at the end of twelve years rece 7311. principal and interest together; what was the delivered him at first?

Ans. £42

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

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