EXAMPLES.

1. The first term of an arithmetical progression is 7, the common difference is ,, and the number of terms is 16. What is the last term ?

1

To solve this we take formula 1 from our table, which is

l=a+(n-1)d. Substituting the above given values for a, d, and n, we find

l=7+1(16-1)=10%. 2. The first term of an arithmetical progression is }, the common difference is 5, and the last térm is 3. What is the number of terms ?

In this example we take formula 13.

8

4

+1;

d which in this present case becomes

31-3

+1=26.

} 3. One hundred stones being placed on the ground in a straight line, at the distance of 2 yards from each other, how far will a person travel who shall bring them one by one to a basket, placed at 2 yards from the first stone ?

In this example a=4; d=4; n=100, which values being substituted in formula 5, give

S=50{8+99x4}=20200 yards, which, divided by 1760, the number of yards in one mile, we get

S=11 miles, 840 yards. 4. What is the sum of n terms of the progression 1, 3, 5, 7, 9, --.- ?

Ans. S=n2.

5. What is the sum of n terms of the progression

1, 2, 3, 4, 5,

Ans. S= n(n+1)

2

GEOMETRICAL RATIO.

(170.) When we compare quantities by seeing how many times greater one is than another, we obtain gcometrical ratio. Thus, the geometrical ratio of 8 to 4 is 2, since 8 is 2 times as great as 4. Again, the geometrical ratio of 15 to 3 is 5.

Equation (2) shows, that in a geometrical ratio the antecedent is equal to the consequent multiplied by the ratio.

Equation (3) shows, that the consequent is equal to the antecedent divided by the ratio.

(171.) When the geometrical ratio of any two terms is the same as the ratio of any other two terms, the four terms together form a geometrical proportion.

a Thus, if-=r; and ira a

(4) ch

¿=r, then will

C

a

с

which relation is a geometrical proportion, and is generally

written thus:

a:c: a'c',

which is read as follows: a is to c, as a' is to c'.

(5)

Of the four quantities which constitute a geometrical proportion, as in arithmetical proportion, the first and fourth are called the extremes, the second and third are called the means.

The first and second constitute the first couplet; the third and fourth constitute the second couplet.

From equation (5), or its equivalent (4), we find

which shows, that the product of the extremes, of a geometri

cal proportion, is equal to the product of the means.

If c=a', then (5) becomes

a: a' a'c',

which changes (6) into

ac'=a'2,

(7)

(8)

so that, if three terms constitute a geometrical proportion, the product of the extremes will equal the square of the mean.

(172.) Quantities are said to be in proportion by inversion, or inversely, when the consequents are taken as antecedents, and the antecedents as consequents.

From (5), or its equivalent (4), which is

Which shows, that if four quantities are in proportion they will be in proportion by inversion.

(173.) Quantities are in proportion by alternation, or alternately, when the antecedents form one of the couplets, and the consequents form the other. Resuming (4), a'

(11) d'

a

с

с

Multiplying both terms of (11) byar

, it will become

a

Ć Therefore, by Art. 171, a:a': :c:c'.

(12) Which shows, that if four quantities are in proportion they will be so by alternation.

(174.) Quantities are in proportion by composition, when the sum of the antecedent and consequent is compared either with antecedent or consequent. Resuming (4),

a'
C

(13)

a

C

=

с

If to (13), we add the terms of the following equation each of whose members is equal to unity, we have

a+c a'+c

ć Therefore, by Art. 171, atc:c::a'tc':c'.

(14) Which shows, that if four quantities are in proportion they will be so by composition.