Ex. 5. Find the sum of the series, 1, 4, 7, 10, 13, &c. to 50 terms. ANS. 3725.

Ex. 6. Find the common difference in an arithmetic series, whose first term is 2, number of terms 4, and sum 16.

Ex. 7. The sum of an arithmetic series is 22, the common

Ex. 8. Find the sum of 50 terms of the series,

Ex. 9. Find the sum of n terms of the series,

Ex. 10. The sum of n terms of an arithmetic series, of which is the first term, is 4n2-3n; find the common difference. ANS. 8.

Ex. 11. If the sum of n terms of an arithmetic series, of which the common difference is 10, be 5n2-4n, what is the first term? ANS. 1.

Ex. 12. How many strokes does a clock strike in 12 hours?

Ex. 13. A gentleman bought a horse upon this condition; that for the first nail upon his shoe he should pay 5 pence; for the second, 8 pence; for the third, 11 pence; and so on for every succeeding nail. Now the number of nails being 32, what did he pay for the horse?

α=

b =

5 Series, 5, 8, 11, &c. to 32 terms.

3

n = 32

S=(10+31.3) 16 103 x 16.

= 1648 pence = £6 17s. 4d.

Ex. 14. A. set out to travel 164 miles, and began by going 10 miles the first day; after this he increased his pace by 3 miles every day. In what time did he arrive at his journey's end?

ANS. 8 days.

Ex. 15. A servant agreed to serve his master for 12 months, upon condition of receiving one shilling for the first month, 3 for the second, 5 for the third, and so on, increasing every month by 2 shillings. Determine the amount of his wages at the expiration of his service,

ANS. £7. 4s.

Ex. 16. A gentleman having bought ten books at different times for £7, found upon examination that the prices of each of them formed a regular series in arithmetical progression, increasing continually by 2 shillings. What did he give for the first?

ANS. 5 shillings.

CHAP. XLI.

OF GEOMETRICAL RATIO.

321. TH HE geometrical ratio between two numbers is determined by resolving the question, How many times one of these numbers is contained in the other? This is found by dividing one number by the other, and the quotient is therefore the ratio sought.

322. We have here three things to consider; 1st. the first of the two numbers proposed, which is termed the antecedent; 2d, the other number, which is called the consequent; 3d, the ratio between them, or the quotient obtained by the division of the antecedent by the consequent. For example, if the ratio between 18 and 12 were required, 18 is the antecedent,

18

12 is the consequent, and 12=14 is the ratio; from which

we determine in this case that the antecedent contains the consequent 14 times.

323. Geometrical ratio is usually represented by two dots placed between the antecedent and consequent, as a : b, expressed a is to' b.

Ratio is therefore expressed by a fraction, of which the antecedent is the numerator, and the consequent the denominator. It is requisite that these fractions, however, should always be reduced to their lowest terms, by dividing them in every case

by the greatest common divisor; thus dividing by 6.

18
12

3

2

is reduced to by

324. Now

324. Now the first species of ratio is the ratio of equality; i. e. when the two terms are equal; as 3: 3, 4 : 4, a : a. Next follow those ratios in which the antecedent is double of the consequent, as 4: 2, which is termed double or duplex ratio; 12: 4, the triple ratio; 16: 4, the quadruple ratio, &c. Those relations in which the consequent contains the antecedent any exact number of times, as 3: 6, 5: 15, &c., are called subduple, subtriple, &c. ratios.

325. If however the consequent is not an exact aliquot part of the antecedent, the value of the ratio can only be expressed by a fraction, or a mixed number, as in Art. 322,

18

the value of the ratio 18: 12 was expressed = 14. If

12

the antecedent be greater than the consequent, it is termed a ratio of greater inequality; and, on the other hand, when the antecedent is less than the consequent, it is called a ratio of lesser inequality.

326. Now since it appears that every ratio may be expressed in the form of a fraction, it is evident that any number of ratios may be compared together, by reducing those fractions to common denominators; for example, if it were required to determine which was the greatest of the two ratios 5 : 4 or

5

9

98, representing them by the fractions and and re

4 8'

ducing them to common denominators, they become

9

8°

Consequently we find that the ratio of 5: 4 is greater

than the ratio of 9: 8.

327. If there be two ratios, the one of greater inequality, and the other of lesser inequality, and the same number be added to each, the former will be diminished and the latter increased. For example, let a +1: a represent a ratio of greater inequality, and let d be added to each of its terms;

the

the new ratio will then be a+1+d:a+d; now the ratio of

the second of which is evidently less than the first, and there. fore the ratio a +1:a has been diminished by adding d to each of its terms. Again, let a-1: a represent a ratio of lesser inequality, and let d be added, and it will become

a-1+d: a+d, and the two fractions

α- 1

a

and

reduced to a common denominator will become,

a2-a+ad-d a2-a+ad

a2 + ad and

a2 + ad

a-1+d

a+d

the second of which is evidently greater than the first, and therefore the ratio of a-1: a has been increased by adding d to each of its terms.

To illustrate this by an example in numbers, let us take the ratio of 5: 4. Adding 1 to each term, it becomes 6: 5.

5

4.

6 25

5 20

24 20'

6

Now and = and of which the second, or is less

5'

than the first; but if, on the contrary, I had been subtracted,

4 16

the ratio would have become = which is greater than 3 12'

5 15

or Therefore a ratio of greater inequality is di4 12'

minished by adding, and increased by subtracting, the same number from each of its terms.

5 25

each of its terms it becomes 5: 6, or

which is

greater