14. Required three numbers in arithmetical progression; the sum of their squares being 1232, and the square of the mean exceeding the product of the extremes by 16.

Ans. 16, 20, and 24. 15. Two men, A and B, set out at the same time; A travels 8 miles a-day; and B travels the first day 1 mile, the second 2, the third 3, &c. In how many days will B overtake A ?

Ans. 15. 16. There is a number, consisting of three places, whose digits are in arithmetical progression ; if it be divided by the sum of its digits, the quotient will be 26; and if 198 be added to the number, the digits will be inverted. What is the number?

Ans. 234. 17. If the sum of six numbers in arithmetical progression be 48 ; and the product of the coinmon difference, multiplied into the least term, be equal to the number of terms; required the common difference and terms.

Ans. d=2;

And 3, 5, 7, 9, 11, and 13 the terms. 18. In a 100 yards taken in a straight line a stone is placed at the end of every yard. What is the least distance that a person must travel from a basket placed at the beginning of ihe first yard to pick up each stone separately and return with it to the basket.

Ans. 5050 yards. 19. To find the 100th term of the senses 7 x 10 x 13 x &c.

Ans. 304. 20. The sum of four whole numbers in arithmetical progression is 10, and the sum of their reciprocals. Required the terms.

Ans. 1, 2, 3,

4. Let 2y=common difference, and 3y=the first or last term.

21. Let the product of five numbers in arithmetical progression be 120, and their sum 15. Required the numbers.

Ans. 1, 2, 3, 4, 5.

Geometrical Progression.

(126.) Definition.- If a series of quantities increase, or decrease, by the continual multiplication or division of the same quantity, then those quantities are said to be in Geometrical progression.

Thus the numbers 1, 2, 4, 8, 16, &c. which increase by the continual multiplication of 2); and the numbers, l, s, , t'q, &c. (which decrease by the continual division of 's, or multiplication of +), are in Geometrical progression.

And thus it appears that each successive term of a geometrical progression may be considered as formed by multiplying the ratio into the last term ; for multiplying by a fraction whose numerator is unity is the same as dividing by the denominator. The general rules are as follows.

(127.) Let a denote the first term of a geometrical progression; r the ratio, or common multiplier ; n the number of terms ; and s the sum of the series;

Then will a +ar+ar? +arst... ar-=s be a general expression for every geometrical progression where the exponents differ b: unity. Multiply both sides of this equation by r and we have

ar+ar+arl +... ara-star" =rs.

From this equation subtract the former and there will remain



that is

a(n-1) = s(r-1); whence s=


Hence, if any three of the quantities, a, ?, n, s, be given, the other may

be found from this equation. COROL. If r be less than 1, and n infinite gum will vanish, therefore the last equation in this case will become sir-1)=-a, or by


changing the signs of all the terms s(1-r)= ai.s=



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Examples. 1. Find the sum of the series 3, 9, 27, &c. to 12 terms. Here a=3,123 and n=12,..

a(r"-1) 3(314-1)


2(531 141-1)


3 X 531440

=3 x 265720=797 160. 2

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Ans. 59049

3. Find three numbers in geometrical progression, such, that their sum shall be equal to 7, and the sum of their squares to 21. Let the numbers be x, y, and z; then by question x+y+z=7,

x+y+ze=21, And by geometrical progression x:y :: Y:Z .. y'=xz, (A). by the first condition r+z=7-y, .. q*+2x2+x=49-147+y®,(B) From this equation subtract iwice equation A and there will remain

14y=28, whence y=2; ..x+%=5 consequently r? + 2xz+z=25, and x +z=17 consequently 23% +27°=34,

subtract the upper from the lower 1–2r2+z=9,
extract the square root

*- %=3

2 + z=5 Whence will be found x=4 and z=1; .. the three numbers are 1, 2, 4. 4. Given a=3, r=5, and n=4, to find s.

468 Ans. 3 The first term of a geometrical series is 2, and the ratio 3 ; required the suin of 20 terms of the series.

3486784400. Ans. 6. The sum of ten terms of the geometrical series 1 1 1 1

9841 1,

1+ Ans. 3° 30° zö' zi , &c. is required.

19683 Miscellaneous Eramples. 1. Required the sum of the series 5, 4, 5, &c. continued for ever.

Ans. 1, 2. What is the sum of 99 terms of the series 1, 2, 4, 8, &c.

Ans. 633825300114114700748351602687.

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3. What is the difference between the 11th and 81st ta rms of the scries in question 2nd ?

Ans. 1208925819614629174705152. 4. What is the sum of the series &c. continued for ever?

Ans. & 5. If the series 1, 3, 9, &c. be carried to 13 terms, how much will the last term exceed the 3d ?

Ans. 531432. 6. What is the sum of the series 9, 6, 4, &c. continued for ever!

Ans. 27. 7. What is the number of acres in an estate, which if sold on the principles of geometrical progression, a farthing being given for the first acre, and a penny for the second, the price of the last acre will be £10,000 ?

Ans. 12,5973l. very nearly. 8. It is agreed to purchase 8 ships of war, of the first rate, on the principles of geometrical progression, the price of the first ship being fixed at 15s, and of the last at £617,657 : 58. What will the second ship cost?

Ans. 5 guineas. 9. Given the sum of three numbers in geometrical progression 39, and the difference between the extremes 24. Required the numbers.

Ans. 3, 9, 27. 10. Supposing ten persons to be living at the end of the first age, ten times as many at the end of the second, &c.; mankind thus dacupling themselves every age. Required the number of persons living at the deluge, supposed to have happened at the end of the sixteenth age.

Ans. 10,000,000,000,000,000. 11. Required four mean proportionals between 5 and 160.

Ans. 10, 20, 40, and 80. 12. What four numbers in geometrical progression are they, whose sum is 24, and the sum of whose squares is 164 ? Ans. 9, 7, 5, 3.

13. Given the sum of the first and second terms of four numbers in a geometrical progression equal to 12, and that of the third and fourth equal to 108. Required the numbers.

Ans. 3,9, 27, 81. 14. The sum of four terms in a geometrical series exceeds the ratio by 1, and the first term is

Required the remaining terms.

Ans. In to, 14. 15. The sum of three numbers in a geometrical progression is 13, and the mean is to the difference between the extremes, as 3 : 8. Required the numbers.

Ans. 1,3,9. 16. Given the difference between the first and second terms of four numbers in a geometrical progression, equal to 54 ; and the difference also between the third and fourth, equal to 6. Required the numbers.

Ans. 81, 27, 9, 3. 17. To find four numbers in geometrical progression whose sum is 15, and the sum of their squares 85 ? Ans. 1, 2, 4, and 8.


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(128.) When a magnitude or number cannot be expressed in fpite terms without the help of a fractional index, it is called a Surd: thus the square root of 2, the cube root of 3, the nth root of a + b, the cube root of (a + x), &c. &c.

may pressed either by 12, 313, ♡a + b, y(a + x)?, &c. or by 27, 35, (a+b), (a+x)}, &c.

Note.--The precise value of these quantities cannot be ascertained; it can only be expressed by means of decimals or series which do not terminate ; and in this sense they are called irrational, to distinguish them from all other quantities wbatever, integral or fractional, whose values are determinate, and which are therefore denominated rational. Surds in their rudicul form, when properly reduced, are subject to all the ordinary Rules of Arithmetic,

The Reduction of Surd quantities.

CASE 1. 129.) A rational quantity may be reduced to the form of a surd, by raising it to the power denoted by the root of he surd.

Example: . 1. Reduce 3 to form of the square root, and it becomes 13° or 79.

3/23 2 2. Reduce

39 3. Reduce a+b...... square root, ....

v(a+b)? 4. Reduce 423 ......cube root, 3646


cule root, ......



(130.) Surd quantities of different indices are reduced to equivalent ones with the same index, by bringing their fractional indices to a common denominator.

Examples. 1. Reduce at and at to surds of the same index.

1 The fractions

and reduced to a common denominator, are

2 3 2

and 6


.: a= at=%a', which are surds with the same and aš= aš=ya!

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aš=%22,) index

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