Examples in Geometrical Progression.

that is, the last term is zero, and the sum is found by dividing the first term by the difference between unity and the ratio.

264. Corollary. From the equation

either of the quantities a, r, and S may be found, when the other two are known.

265. EXAMPLES.

1. Find the sum of the infinite progression, of which the first term is 1, and the ratio.

Ans. 2.

2. Find the sum of the infinite progression of which the first term is 0.7, and the ratio 0.1.

Ans. f.

3. Find r, in an infinite progression, when a and S are known.

4. Find the ratio of an infinite progression, of which the first term is 17, and the sum 18.

Ans.

5. Find a, in an infinite progression, when r and S are known.

Ans. a S(1 —r).

6. Find the first term of an infinite progression, of which the ratio is, and the sum 6.

Ans. 2.

Form of any Equation.

CHAPTER VIII.

GENERAL THEORY OF EQUATIONS.

SECTION I.

Composition of Equations.

266. Any equation of the nth degree, with one unknown quantity, when reduced as in art. 118, may be represented by the form

n-1

n

A x2 +В x2-1+C xn−2 +&c. + M = 0.

If this equation is divided by A, and the coefficients B C M

&c., represented by a, b, &c., m, it is reduced to A

x"+ax"-1+bxn−2+ &c. + m = 0.

267. Theorem. If any root of the equation

n

x2 + a xn−1 + bxn−2+ &c. + m = 0

is denoted by x', the first member of this equation is a polynomial, divisible by x-x', without regard to the value of x.

Proof. Denote x-x' by x1, that is,

If this value of x is substituted in the given equation, if Pz is used to denote all the terms multiplied by x[1), or

Form of any Equation.

by any power of 11, and Q used to denote the remaining terms, the equation becomes

P+Q= 0.

Now the given equation is, by hypothesis, satisfied by the value of x.

The terms not multiplied by x1, or a power of x[1], must, therefore, cancel each other; and the first member of the given equation becomes

P x1,

which is divisible by xl, or its equal x

268. Corollary. The preceding division is easily effected by subtracting from the polynomial

-2

x2 + a x2-1 + bxn−2 + &c. +m,

the expression

x'n+ax'n-1+ bx' n −2+&c. + m = 0,

which does not affect its value, but brings it to the form

-2

x-x'"+a(x-1-x'n-1)+b (xn−2 — x'n-2)+&c.,

of which each term is, by art. 49, divisible by x-x'. The quotient is, by art. 50,

269. Corollary. The first term of the preceding quotient is x-1; and if the coefficients of x^-2, x^-3, &c., are denoted by a', b', &c., the quotient is

Number of the Roots of an Equation.

and the equation of art. 267, is

-1

(x − x') (xn−1 + a' xn−2 + b' xn−3+ &c.) = 0; which is satisfied either by the value of x,

x = x',

or by the roots of the equation

xn−1 + a' xn−2+ b' x2 - 3 + &c. = 0.

If now " is one of the roots of this last equation, we have in the same way

xn−1 + a'xn−2+&c. = (x − x') (xn−2+ a′′xn−3+&c.) = 0, and the given equation becomes

-2

(x — x') (x — x'') (xn−2+a" x2-3+ &c.) = 0; which is satisfied by the value of x",

x= x";

so that " is a root of the given equation.

By proceeding in this way to find the roots x"", x, &c., the given equation may be reduced to the form

(x — x') ( x − x") (x — x'") (x — x1) &c = 0,

is the same with the degree n of the given equation; and, therefore, the number of roots of an equation is denoted by the degree of the equation; that is, an equation of the third degree has three roots, one of the fourth degree has four roots, &c. But all these roots are not necessarily real or rational; they may, on the contrary, be irrational or even imaginary.

270. Scholium. Some of the roots x', x'', x''', &c., are often equal to each other, and in this case the number of unequal roots is less than the degree of the equation.

Imaginary Roots.

Thus the number of unequal roots of the equation of the 9th degree,

(x —7) (x+4)3 (x — 1)5 — 0,

is but three, namely, 7, —4, and 1, and yet it is to be regarded as having 9 roots, one equal to 7, three equal to - -4, and five equal to 1.

271. Corollary. The equation

would appear to have but one root, that is,

but it must, by art. 269, have n roots, or rather, the nth root of a must have n different values.

must be divisible by z — -1, and we have

x3 — 1 = (x − 1) (x2 + x + 1) = 0.

Now the roots of the equation

x=}(−1+v−3), and — § (— 1 —√−3).

=