THEOREM 5.-If four quantities, 2, 6, 4, 12, are di

8. By Divifion,

:

:: 4: 12

r

Because the product of the means in each cafe, is equal to that of the extremes, and therefore the quantities are proportional by Theorem 2.

THEOREM 6.—If three numbers, 2, 4, 8, be in continued proportion, the square of the firft will be to that of the fecond, as the firft number to the third; that is, 2 X 2:4 X 42: 8.

THEOREM 7-In any continued geometrical proportion, (1, 3, 9, 27, 81, &c) the product of the two extremes, and that of every other two terms, equally diftant from them, are equal.

THEOREM 8. The fum of any number of quantities, in continued geometrical proportion, is equal to the dif ference of the rectangle of the second and last terms, and the fquare of the first divided by the difference of the firft and fecond terms.

GEOMETRICAL PROGRESSION.

A Geometrical Progreffion is when a Rank or Series, of numbers, increases, or decreases, by the continual multiplication, or divifion, of fome equal number.

PROBLEM I. Given one of the extremes, the ratio, and the number of the terms of a Geometrical Series, to find the other extreme.

RULE.-Multiply, or divide (as the cafe may require) the given extreme by fuch power of the ratio, whofe ex

ponent

ponent* is equal to the number of terms lefs 1, and the product, or quotient, will be the other extreme.

EXAMPLES.

1. If the first term be 4, the ratio 4, and the number of terms 9; what is the last term?

1. 2. 3. 4. + 4= 8

4. 16. 64. 256. × 256=65536=Power of the ratio, whofe exponent is lefs by 1, than the number of terms. 65536 8th power of the ratio.

Multiply by 4-firft term.

262144 last term.

Or, 4848=262144 the Anf.

2. If As the laft term, or any term near the laft, is very tedious to be found by continual multiplication it will often be neceffary,in order to afcertain them, to have a series of numbers in Arithmetical Proportion, called Indices, or Exponents, beginning either with a cypher or an unit, whose common difference is one.

When the firft term of the feries and the ratio, are equal, the indices must begin with a unit, and, in this cafe, the product of any two terms is equal to that term fignified by the sum of their indices. 1.2. 3. 4. 5. 6. & Indices or Arithmetical feries. Thus, 24.8.16.32.64 &c. Geometrical feries (leading terms.) Now, 6+6=12=the index of the twelfth term, and

64 X 64 4096 the twelfth term.

But when the firft term of the feries and the ratio are different, the indices must begin with a cypher, and the fum of the indices, made choice of, must be one lefs than the number of terms, given in the queftion; because I in the indices ftands over the fecond term, and 2 in the indices over the third term, &c. And, in this cafe, the product of any two terms, divided by the firft, is equal to that term beyond the first, fignified by the fum of their indices.

Thus {

O I. 2.

3. 4. 5. 6, &c. Indices. 1.3. 9. 27. 81. 243. 729, &c. Geometrical feries. Here, 6 + 5. 11 the Index of the 12th term. 729×243=177147 the 12th term,because the first term of the feries and the ratio are different, by which mean a cypher ftands over the first term,

Thus, by the help of these indises, and a few of the first terms in any geometrical feries, any term, whofe distance from the first term is affigned, though it were ever so remote, may be obtained without producing all the terms.

2. If the laft term be 262144, the ratio, 4, number of terms 9; what is the first term? Sth power of the ratio 4°=65536)262144

and the

the first (term.

Again, Given the first term, and the ratio to find any other term affigned.

RULE 1.-When the indices begin with an unit.

1. Write down a few of the leading terms of the feries, and place their indices over them.

2. Add together fuch indices, whose fum fhall make up the entire index to the term required.

3. Multiply the terms of the geometrical feries, belonging to thofe indices, together, and the product will be the term fought

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

2. A merchant wanting to purchase a cargo of horses for the Weftindies, a jockey told him he would take all the trouble and expenfe, upon himself, of collecting and purchafing 30 horfes for the voyage, if he would give him what the last horfe would come to by doubling the whole number by a half penny, that is, two farthings for the firft, four for the second, eight for the third, &c. to which the merchant, thinking he had made a very good bargain, readily agreed: Pray, what did the last horfe

come

Note. If the ratio of any geometrical series be double, the difference of the greatest and leaf terms is equal to the fum of all the terms, except the greatest: If the ratio be triple, the difference is double the fum of all but the greatest: If the ratio be quadruple, the difference is triple the sum of all but the greatest &c

In any geometrical feries decreafing, and continued od in. finitum, balf the greatest term is equal to the sum of all the remaining terms, ad infinitum.

come to; and, what did the horses, one with another, coft the merchant ?

[30th, or laft term. 1. 2. 3. 4. 5. 6+ 6= 12th. 12 + 12 + 6= 2. 4. 8. 16. 32. 64 × 64=4096&4096 × 4096 × 14 1073741824 grs.£1118481 15. 4d. and their average price was 37232 14s. Od. a piece.

RULE 2.When the indices begin with a cypher. 1. Write down a few of the leading terms of the feries, as before, and place their indices over them.

2. Add together the most convenient indices to make an Index, lefs by 1 than the number expreffing the place of the term fought.

3 Multiply the terms of the geometrical feries, togeth er, belonging to thofe indices, and make the product a dividend.

4. Raife the first term to a power whofe index is one lefs than the number of terms multiplied, and make the refult a divifor, by which divide the dividend, and the quotient will be that term beyond the firft, fignified by the fam of thofe indices, or the term fought.

5. If the first term be 5, and the ratio 3, What is the 7th term.

O. I. 2. 3+2+1=

6 Ind. to the term be (yond ift, or 7th.

5. 15. 45 135X45X15=91125-Dividend. The number of terms, multiplied. is 3 (viz. 135×45 ×15) and 3-1=2, is the power to which the term 5 is to be raised; but the zd power of 5 is 5 x525. therefore 91125÷25-3545 the 7th term required.

and

PROB. 2. Given the firft term, the ratio, and number of terms, to find the fum of the feries.

RULE. Raife the ratio to a power, whofe Index fhall be equal to the number of terms. from which fubtract I; divide the remainder by the ratio, less

1, and the quotient multiplied by the first term, will give the fum of the feries.

EXAMPLES.

1. If the firft term be 5, the ratio 3, and the number What is the fum of the feries?

of terms 7;

Ratio=3X3X3X3X3X3×3=2187=7th power

Subtract

I (of the ratio.

2. A fhopkeeper fold 13 yards of cloth on the following terms, viz zd. for the first yard, 4d for the fecond, 8d. for the third, &c. I demand the price of the cloth.

21

2

13

Σ

X2=16382d.681 5s 2d. Anf.

3. A gentleman whofe daughter was married on a new year's day, gave her guinea, promifing to triple it on the first day of each month in the year? Pray what did her portion amount to ?

3X1=265120 guineas, Anf.

3-1

n

4. What debt can be difcharged in a year, by paying 'hilling the first month, 10s. the fecond, and fo on, each month a tenfold proportion? -O -I X !== {{{111111 fhillings, $55555555551. 11s. Anf

Y

5. A