tor and denominator, (Art. 128), will become

ar'(1

will be indefinitely small

when n is indefinitely great; and consequently, by prolong

ar'

ing the series, S may be made to differ from by less than

any assignable quantity.

488. Whence, supposing the series to be continued indefinitely, or without end, we shall have in that case, S=

ar

which last expression is what some call the radix, and others the limit of the series; as being of such a value, that the sum of any number of its terms, however great, can never exceed it, and yet may be made to approach nearer to it than by any given difference.

489. If the ratio, or multiplier, r, be negative, in which case the series will be of the form

1

And if r be a proper fraction, as before, we shall have,

for the sum of an indefinite number of terms of the series a

Ex. 1. Find the sum of the series, 1, 3, 9, 27, &c. to 12 terms.

Ex. 2. Find three geometric means between 2 and 32.

Here a=2,

1=32,

m=3;

and the means required are 4, 8, 16.

Ex. 3. The first term of a geometrical progression is 1, the ratio 2, and the number of terms 10. What is the sum of the series? Ans. 1023.

Ex 4. In a geometrical progression is given the greatest term=1458, the ratio =3, and the number of terms find the least term.

7, to

Ans. 2.

Ex. 5. It is required to find two geometrical proportionals between 3 and 24, and four geometrical means between 3 and Ans. 6 and 12; and 6, 12, 24, and 48. Ex. 6. Find two geometric means between 4 and 256.

96.

Ans. 16, and 64.

Ex. 7. Find three geometric means between and 9.
Ans. 1, 1, 3.

Ex. 8. A Gentleman who had a daughter married on Newyear's day, gave the husband towards her portion 4 dollars, promising to triple that sum the first day of every month, for nine months after the marriage; the sum paid on the first day of the ninth month was 26244 dollars. What was the Lady's fortune? Ans. 39364 dollars.

Ex. 9. Find the value of 1+1+1+1+&c. ad infinitum.

Ans. 2.

Ex. 10. Find the value of 1++++ &c. ad infini

tum.

Ans. 4.

III. HARMONICAL PROPORTION AND PROGRESSION.

490. Three quantities are said to be in harmonical proportion, when the first is to the third, as the difference between the first and second is to the difference between the second and third.

Thus, a, b, c, are harmonically proportional, when

a:c::a-b: b—c, or a : c : : b—a : c—b. And c, [since a(b—c)=c(a—b) or ab=(2a−b)c], is a third ab

harmonical proportion to a and b, when c=

2a-b

491. Four quantities are in harmonical proportion, when the first is to the fourth, as the difference between the first and second is to the difference between the third and fourth. Thus, a, b, c, d, are in harmonical proportion, when

a:d :: a-b: c-d, or a : d :: b—a : d—c.

And d, [since a(c-d)=d(a-b) or ac=(2a-b)d], is a fourth harmonical proportional to a, b, c, when d

ae

2a-b

In each of which cases, it is obvious, that twice the first term must be greater than the second, or otherwise the proportionality will not subsist.

492. Any number of quantities, a, b, c, d, e, &c. are in harmonical progression, if a : c :: a-b : b- -C -d; ce::c-d: d-e, &c.

bd::bc

493. The reciprocal of quantities in harmonical progression, are in arithmetical progression. For, if a, b, c, d, e, &c. are in harmonical progression; then, from the preceding Article, we shall have bc+ab=2ac; dcbc2db; ed+cd=2ec, &c. Now, by dividing the first of these equalities by abc;

the second by bdc; the third by cle; &c., we have, +

2 1 1 2 1 1 2

+

;

+==; &c. Therefore, (Art. 464). b'b d C C e d

a

C 1 11

494. An harmonical mean between any two quantities, is equal to twice their product divided by their sum. For if a. x, b, are three quantities in harmonical proportion, then (Art. 490), 2ab. a : b ;; a—x : x—b ; .. ax—ab=ab—bx, and x=a+b

Ex. 1. Find a third harmonical proportional to 6 and 4. Let x= the required number, then 6 :x :: 6-4 ; 4—x ; .. 24-6x=2x, and x=3.

Ex. 2. Find an harmonical mean between 12 and 6.

Aus. 8. Ex. 3. Find a third harmonical proportional to 234 and Ans. 104.

144.

3.

Ex. 5. Find a fourth harmonical proportional to 16, 8, and

Ans. 2.

IV. PROBLEMS IN PROPORTION AND PROGRESSION.

Prob. 1. There are two numbers whose product is 24, and the difference of their cubes: cube of their difference : : 19: 1. What are the numbers?

Let x= the greater number, and y= the lesser.

Then, xy=24, and 3-y3: (xy)3 :: 19: 1. By expansion, x3-y3 : x3-3x2y+3xy3—y3 :: 19 : 1 ; .. (Art. 480), 3x2y-3ry: (x-y): 18:1; and, (Art. 481), dividing by x-y, 3xy: (x-y)3 :: 18:1 ; but xy=24;.. 72: (x-y): 18: 1. Hence, (Art. 190), 18(x—y)2=72, or(x—y)2=4 ;

Prob. 2. Before noon, a clock which is too fast, and points to afternoon time, is put back five hours and forty minutes ; and it is observed that the time before shown is to the true time as 29 to 105. Required the true time.

Ans. 8 hours, 45 minutes. Prob. 3. Find two numbers, the greater of which shall be to the less as their sum to 42, and as their difference to 6. Ans. 32, and 24.

Prob. 4. What two numbers are those, whose difference; sum, and product, are as the numbers 2, 3, and 5, respectively? Ans. 10, and 2. Prob. 5. In a court there are two square grass-plots; a side of one of which is 10 yards longer than the other; and their areas are as 25 to 9. What are the lengths of the sides? Ans. 25, and 15 yards.

Prob. 6. There are three numbers in arithmetical progression, whose sum is 21; and the sum of the first and second is to the sum of the second and third as 3 to 4. Required the numbers.

Ans. 5, 7, 9, Prob. 7. The arithmetical mean of two numbers exceeds the geometrical mean by 13, and the geometrical mean exceeds the harmonical mean by 12. What are the numbers? Ans. 234, and 104. Prob. 8. Given the sum of three numbers, in harmonical proportion, equal to 26, and their continual product =576 ; to find the numbers.

Ans. 12, 8 and 6. Prob. 9. It is required to find six numbers in geometrical progression, such, that their sum shall be 315, and the sum of the two extremes 165.

Ans. 5, 10, 20, 40, 80, and 160.

Prob. 10. A number consisting of three digits which are in arithmetical progression, being divided by the sum of its digits, gives a quotient 48; and if 198 be subtracted from it, the digits will be inverted. Required the number.

.

Ans. 432. Prob. 11. The difference between the first and second of four numbers in geometrical progression is 36, and the diffe

rence between the third and fourth is 4; What are the numbers ?

Ans. 54, 18, 6, and 2. Prob. 12. There are three numbers in geometrical progression; the sum of the first and second of which is 9, and the sum of the first and third is 15. Required the numbers. Ans. 3, 6, 12.

Prob. 13. There are three numbers in geometrical progression, whose continued product is 64, and the sum of their cubes is 584. What are the numbers ?

Ans. 2, 4, 8. Prob. 14. There are four numbers in geometrical progression, the second of which is less than the fourth by 24; and the sum of the extremes is to the sum of the means as 7 to 3. Required the numbers.

Ans. 1, 3, 9, 27. Prob. 15. There are four numbers in arithmetical progression, whose sum is 28; and their continued product is 585. Required the numbers.

Ans. 1. 5, 9, 13.

Prob. 16. There are four numbers in arithmetical progression; the sum of the squares of the first and second is 34; and the sum of the squares of the third and fourth is Required the numbers.

130.

Ans. 3, 5, 7, 9.