(193.) Equal multiples of two quantities have the same ratio as the quantities themselves.

a

ō

The ratio of a to b is represented by the fraction

and the value of a fraction is not changed if we

multiply or divide both numerator and denominator by the same quantity. Thus,

(194.) If there is any number of proportional quantities all having the same ratio, the first will have to the second the same ratio that the sum of all the antecedents has to the sum of all the consequents.

Let a, b, c, d, e, ƒ be any number of proportional quantities, such that

then will For, since

we have

And since

we have

a:b::c:d::e:f,
a:b::a+c+e:b+d+f.
a:b::c:d,

ad=bc.

a:b::e:f,

-af-be.

QUEST.-Compare equal multiples of two quantities. What principle may be applied to any number of proportional quantities all having ne same ratio?

To these equals add ab=ba,

and we obtain a(b+d+f)=b(a+c+e). Hence, by Art. 184,

(195.) If four quantities are proportional, like powers or roots of these quantities will also be proportional.

and the same may be proved of the cube or any other power.

(196.) If there are two sets of proportional quantities, the products of the corresponding terms will be proportional.

QUEST.-Compare like powers or roots of proportional quantities. What principle may be applied to two sets of proportional quantities?

SECTION XIV.

PROGRESSIONS.

ARITHMETICAL PROGRESSION.

(197.) An Arithmetical Progression is a series of quantities which increase or decrease by the continued addition or subtraction of the same quantity.

Thus the numbers

1, 3, 5, 7, 9, 11, etc.,

which are obtained by the addition of 2 to each successive term, form what is called an increasing Arithmetical Progression; and the numbers

20, 17, 14, 11, 8, 5, etc.,

which are obtained by the subtraction of 3 from each successive term, form what is called a decreasing Arithmetical Progression.

PROBLEM I.

(198). To find any term of an Arithmetical Progression.

If a represent the first term of an increasing arithmetical progression, and d the common difference, the second term of the series will be a+d, the third a+2d, the fourth a+3d, the fifth a+4d, etc.

QUEST. What is an Arithmetical Progression? What is an increasing Progression? What is a decreasing Progression? How may we find any term of an Arithmetical Progression?

The coefficient of d in the second term is 1, in the third term 2, in the fourth term 3, and so on; that is, any term of the series is equal to the first term, plus as many times the common difference as there are preceding terms.

If we represent any term of the series by l, and suppose n to be the number which marks the place of that term in the series, the expression for this term will be

l=a+(n-1)d.

(199.) Hence, if we put / to represent the last term of the series, we shall have the following

RULE.

The last term of an increasing arithmetical progression is equal to the first term, plus the produc of the common difference into the number of terms less one.

This rule enables us to find any term of a series without being obliged to determine all those which precede it.

Examples.

1. What is the fourth term of a series whose first term is 3 and common difference 2?

Ans. 9.

2. What is the sixth term of a series whose first

term is 5 and common difference 3?

Ans. 20.

QUE Give the rule for finding the last term of an increasing