of the partial progressions; and since the last term of the first, forms the first term of the second, &c., we may conclude that all of these partial progressions form a single progression.

GENERAL EXAMPLES.

1. Find the sum of the first fifty terms of the progression 2.9.16.23

3. Find the sum of 100 terms of the series 1.3.5.7.9... Ans. 10000.

4. The greatest term considered is 70, the common difference 3, and the number of terms 21: what is the least term and the sum of the terms?

Ans. Least term 10; sum of terms 840.

5. The first term of a decreasing arithmetical progression is 10, the common difference is , and the number of terms

21 required the sum of the terms.

Ans. 140.

6. In a progression by differences, having given the common difference 6, the last term 185, and the sum of the terms 2945: find the first term, and the number of terms.

7. Find 9 arithmetical means between each antecedent and consequent of the progression 2.5.8. 11. 14 . . .

Ans. d=0.3.

8. Find the number of men contained in a triangular bat talion, the first rank containing 1 man, the second 2, the third 3, and so on to the nth, which contains n. In other words,

find the expression for the sum of the natural numbers 1, 2,

9. Find the sum of the first n terms of the progression of uneven numbers 1, 3, 5, 7%, 9 Ans. S no.

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

Ans. 11 miles 840 yards.

Of Ratio and Geometrical Proportion.

181. The RATIO of one quantity to another, is the quotient which arises from dividing the second by the first. Thus, the

182. Two quantities are said to be proportional, or in proportion, when their ratio remains the same, while the quantities themselves undergo changes of value. Thus, if the ratio of a to b remains the same, while a and b undergo changes of value, then a is said to be proportional to b.

183. Four quantities are in proportion, when the ratio of the first to the second, is equal to the ratio of the third to the fourth.

the quantities a, b, c and d, are said to be in proportion. We generally express that these quantities are proportional by writing them as follows:

a : b : : c: d.

This algebraic expression is read, a is to b, as c is to d, and is called a proportion.

184. The quantities compared, are called terms of the proportion.

The first and fourth terms are called the extremes, the second and third are called the means; the first and third are called antecedents, the second and fourth are called consequents, and the fourth is said to be a fourth proportional to the other three.

If the second and third terms are the same, either of these is said to be a mean proportional between the other two. Thus, in the proportion

b is a mean proportional between a and c, and c is said to be a third proportional to a and b.

185. Two quantities are reciprocally proportional when one is proportional to the reciprocal of the other.

Geometrical Progression.

186. A GEOMETRICAL PROGRESSION is a series of terms, each of which is derived from the preceding one, by multiplying it by a constant quantity, called the ratio of the progression.

If the ratio is greater than 1, each term is greater than the preceding one, and the progression is said to be increasing. If the ratio is less than 1, each term is less than the preceding one, and the progression is said to be decreasing.

Thus,

3, 6, 12, 24, . . . &c., is an increasing progression.

It may be observed that a geometrical progression is a continued proportion in which each term is a mean proportional between the preceding and succeeding terms.

187. Let r designate the ratio of a geometrical progression,

We deduce from the definition of a progression the follow ing equations:

and, in general, any term n, that is, one which has n 1 terms before it, is expressed by art.

Let be this term; we have the formula

by means of which we can obtain any term without being obliged to find all the terms which precede it. That is,

Any term of a geometrical progression is equal to the first term multiplied by the ratio raised to a power whose exponent denotes the number of preceding terms.

EXAMPLES.

1. Find the 5th term of the progression

2:48 16, &c.,

in which the first term is 2, and the common ratio 2.

188. We will now explain the method of determining the sum of n terms of the progression

of which the ratio is r.

If we denote the sum of the series by S, and the nth term by, we shall have

and by subtracting the first equation from the second, member

To obtain the sum of any number of terms of a progression, by quotients,

Multiply the last term by the ratio, subtract the first term from this product, and divide the remainder by the ratio diminished by 1.

EXAMPLES.

1. Find the sum of eight terms of the progression

2. Find the sum of five terms of the progression

3. Find the sum of ten terms of the progression

4. What debt may be discharged in a year, or twelve months, by paying $1 the first month, $2 the second month, $4 the third month, and so on, each succeeding payment being double the last; and what will be the last payment?

Ans. Debt, $4095; last payment, $2048.

5. A gentleman married his daughter on New-Year's day, and gave her husband 1s. toward her portion, and was to double it on the first day of every month during the year: what was her portion? Ans. £204 15s.