Prop. 3. In any series of numbers in geometrical progression, decreasing, ad infinitum,-given the first term and the ratio to find the sum of the series.

Rule. Subtract the second term from the first; the square of the first term, divided by this difference, will give the sum of the series.

See the 7th note in circulating decimals, Part I. page 105.

Note. If the least term,

g=the greatest,

n the number of terms.

s the sum of the terms,

r=the ratio,

log logarithm of any letter.

Then will the following theorems exhibit all the possible cases of geometrical progression, including those already given.

Proposition 1. Given l, g, and n, to find s and r.

and the value of r in the second, must be found as directed in the 2d nore in Double Position.

Where the value of l in the first equation, and the value of r in the second, must be found as directed in the second note in Double Position, being the most difficult proposition in geometrical progression,

The above theorems will answer for any finite series of numbers, either increasing or decreasing; (see note 9th Arithmetical Progression.) But, if the series decrease, ad infinitum, then n will be infinite, or greater than any assignable number, and 10. Hence the three following theorems answer all the possible cases of an infinitely decreasing geometrical progression.

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ratio be a fraction, then r must represent the reciprocal of that frac tion. Thus if the ratio be 3, then, &c.

Examples to Proposition 1.

(1.) The first, or least, term of a series of numbers in geometrical progression is 3, the ratio 3, and the number of terms 14, what is the greatest, or last term?

5

+

27 81
•

243, &c. geometrical scries.

5313, an unit less than the number of terms 243 X 243 x 27: 1594323

Then 1594323 x 34782969 the 14th term.

(2.) If the first, or least, term be 2, the ratio 2, and the number of terms 19, what is the last, or greatest, term? (3.) A draper sold 20 yards of cloth; the first yard for 3d., the second for 9d., the third for 27d., &c. in triple proportion geometrical; what did he sell the last yard for? (4.) The first, or least, term of a geometrical series is 5, the ratio 3, and the number of terms 12; what is the last, or greatest, term?

3 9 27 81. 243, &c. geometrical series

3

+

4+4

27 x 81 X 81

Then 177147 x 5 =

11 one less than the number of terms
177147

885735, answer.

Note. If the greatest term (885735) had been given to find the least (5), the operation would have been the same, excepting that 885735 must have been divided by 177147.

(5.) If the first, or least, term be 7, the ratio 2, and the number of terms 19, what is the last, or greatest, term?

(6.) A thrasher worked 20 days for a farmer, and received (by agreement) for the first day's work 4 barleycorns, for the second 12, for the third 36, &c., in triple proportion geometrical; what did he receive for his last day's work, admitting 7680 barley-corns to fill a pint measure, and the barley to be worth 2s. 6d. per bushel?

Examples to Prop. 2

(7.) If the first term of a series of numbers in geometrical progression be 5, the ratio 3, and number of terms 12, what is the sum of the terms?

The last or greatest term (by example 4,) is 885735.

Then 885735-5-885730 difference between the extremes. And 3-1-2 ratio less 1. Hence 885730÷2=442865; and 412865+885755=1328600, sum of the terms.

(8.) If the first term be 4, the ratio 3, and the number of terms 7, what is the sum of the terms?

CLASS II. exercising all the preceding propositious.

(9.) What would a horse be sold for that has 4 shoes on, with 8 nails in each shoe, at 1 farthing for the first nai!, 2 for the second, 4 for the third, &c. And, supposing another horse to be sold with only two shoes on, on the same conditions, what would be the difference in their prices?

(10.) If a servant should agree with his master to serve him 11 years, without any other reward than the produce of a wheat-corn for the first year; and, for the second year, ground sufficient to sow his first year's produce on, &c. from year to year to the end of the time: what would his wages amount to, admitting each wheat-corn to yield ten by sowing, 7680 wheat-corns to fill a pintmeasure, and that he could sell his wheat at 8s. per bushel?

(11.) A nobleman dying, left ten sons, to whom and to his executor he bequeathed his estate as follows: to his executor he gave 10247., the youngest son was to have as much and half as much, and every son to succeed the next younger in the same ratio of 1; what was the eldest son's fortune, and what did the nobleman die worth?

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(12.) Required the sum of 1, 3, 3, 27, &c. continued 15 terms?

(13.) Required the sum of, 30, Toto, Todoo, &c. carried to 12 terms.

(14.) The greater extreme of a descending series in geometrical progression is 1835008, the ratio 2, and the number of terms 19; what is the sum of the terms?

Examples to Prop. 3.

(15.) Required the sum of +180+TOGO+To3oo, &c. ad infinitum.

difference between the first and second terms. square of the first term. Then, answer. Hence we may infer, that if a ball were put in motion by a force, which moved it of a league, or 1584 yards, the first minute, (or any portion of time,) of a league, or 1583 yards, the second, &c. for over, it would go no farther than 1 mile! For, it is evident, that ,&c. ad infinitum, =3333, &c. ad infinitum; and this is equal to } precisely, by the nature of vulgar fractions and infinite decimals.

(16.) Required the sum of {+1+}+7%, &c. ad infinitum.

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(17.) Required the sum of 3+3+27+37, &c. ad infinitum.

(18.) If a body be put in motion by a force which moves it 10 miles the first portion of time, 9 miles in the second equal portion, and so on (in the ratio of 2) for ever, how many miles will it pass over?

VARIATIONS.

Definition. By Variations are meant, the different ways any number of things may be altered, or changed, with respect to their places. These are sometimes called Changes, Permutation, Alternation, &c.

Proposition 1. To find the number of changes that can be made of any given number of things, all different from each other.

Rule. Multiply continually together the numbers 1, 2, 3, 4, 5, &c., to the number of terms; and the last product will be the answer.