term will be equal to the number of terms that precede that term.' Therefore in the laft term * you must have nx b, fo that x must be equal to a + n − 1 × b. And the sum of all

the terms being a + xx, it will also be equal

2an + n2 b-nb

n b

to

or to a +

2

2

b

xn. Thus for example, the feries 1+ 2+ 3+ 4+ 5 &c. continued to a hundred, must be equal to

§ 63. If a feries have (o) nothing for its first term, then "its fum fhall be equal to half the product of the last term multiplied by the number of terms." For then, a being = 0, the fum of the terms, which is in general a + x ×

n

From which it is evi

dent, that "the fum of any number of arithmetical proportionals beginning from nothing, is equal to half the fum of as many terms equal to the greatest term."

Thuso + 1+ 2+ 3+ 4+ 5+ 6+ 7+ 8+ 9 = 9+9+9+9+9+9+9+9+9+9 _ 10×9 = 45.

2

2

$64. If of four quantities the quotient of the first and fecond be equal to the quotient of the third and fourth, then thofe quantities are

faid to be in Geometrical proportion. Such are the numbers 2, 6, 4, 12; and the quantities a, ar, b, br; which are expreffed after this

manner;

2 64 12.

a: ar:: b: br.

And you read them by faying, As 2 is to 6, fo is. 4 to 12; or as a is to ar, so is b to br.

In four quantities geometrically proportional, "the product of the extremes is equal to the product of the middle terms." Thus a × br = arx b. And, if it is required to find a fourth proportional to any three given quantities,

ઃઃ

multiply the fecond by the third, and divide the product by the first, the quotient shall give the fourth proportional required." Thus, to find a fourth proportional to a, ar, and b, I multiply ar by b, and divide the product ar b by the first term a, the quotient br is the fourth proportional required.

$65. In calculations it fometimes requires a little care to place the terms in due order; for which you may obferve the following Rule.

First fet down the quantity that is of the fame kind with the quantity fought, then confider, from the nature of the queftion, whether that which is given is greater or less than that which is fought; if it is greater, then place the greatest of the other two quantities on the left hand; but if it is lefs, place the leaft of the other two quantities on the left hand, and the other on the right." Then

Then fhall the terms be in due order; and you are to proceed according to the rule, multiplying the fecond by the third, and dividing their product by the first.

EXAMPLE.

If 30 men do any piece of work in 12 days, bow many men fhall do it in 18 days?

Because it is a number of men that is fought, firft fet down 30, the number of men that is given: I easily fee that the number that is given is greater than the number that is fought, therefore I place 18 on the the left hand, and 12 on the right; and find a fourth proportional to 18, 30, 12, viz.

30 X 12

18

= 20.

$66. When a series of quantities increase by one common multiplicator, or decrease by one common divifor, they are faid to be in "Geometrical proportion continued."

As a, ar, ar2, ar3, art, ar3, &c;

The common multiplier or divifor is called. their "

common ratio."

In fuch a feries, "the product of the first and laft is always equal to the product of the Second and last but one, or to the product of any two terms equally remote from the extremes." In the feries a, ar, ar, ar3, &c. if y be the laft term,

then

then fhall the four laft terms of the series be

y y

; now it is plain that a x y = ar x

2/2 = ar2 × 2/2 = ar3 × 2, &c.

$67. "The fum of a series of geometrical proportionals wanting the first term, is equal to the fum of all but the laft term multiplied by the common ratio."

For ar + ar2 + ar3 &c. +/+ 2/2 +

y

+

+ y

= r × a + ar + ar2 &c. + + + 2/2 + Therefore if s be the fum of the feries, sa will be equal to sy x r; that is s — a = sr-yr, or srs = yra, and s=

yr

$68. Since the exponent of r is always increafing from the fecond term, if the number of terms ben, in the last term its exponent will be n - I. Therefore y = ar"-1; and yr = ar" -1+1 = ar"; and s =

7

ara. So

==

that having the firft term of the feries, the number of the terms, and the common ratio, you may easily find the fum of all the terms.

If it is a decreafing feries whofe fum is to be found, as of y + 2 + 2/2 + 2/3 &c. + ar3 + ar2 + ar + a, and the number of the terms be

See the Rules in the following Chapter.

fup

fuppofed infinite, then shall a, the last term, be equal to nothing. For, because n, and confe

quently is infinite, a =

yr

= o. The fum

of fuch a feries s = ; which is a finite fum,

though the number of the terms be infinite.

Thus 1 + + + { + π + &c. =

6

and 1 + + + + + &c. =

2- I

I X 2

= 2.

СНАР. Х.

OF EQUATIONS THAT INVOLVE ONLY ONE UNKNOWN QUANTITY.

§ 69. AN equation is « a propofition alerting

the equality of two quantities." It is

expreffed most commonly by fetting down the quantities, and placing the fign (=) between them.

An equation gives the value of a quantity, when that quantity is alone on one fide of the equation: and that value is known, if all those that are on the other fide are known. Thus if 4 × 6 I find that x=

3

= 8, I have a known va

lue of x. These are the laft conclufions we are

to