Hence the sum of an infinite series decreasing in geometrical progression is found by the following

RULE.

Divide the first term by unity diminished by the ratio.
Ex. 1. Find the sum of the infinite series

Ex. 2. Find the sum of the infinite series

1++++, &c.

Ex. 3. Find the sum of the infinite series

Ans.

Ex. 4. Find the ratio of an infinite progression, whose first term is 1, and the sum of the series.

Ans. .

Ex. 5. Find the first term of an infinite progression, whose ratio is, and the sum 3.

Ans.

Ex. 6. Find the first term of an infinite progression, of which

(246.) Prob. 1. Of four numbers in geometrical progression, the sum of the first and second is 15, and the sum of the third and fourth is 60. Required the numbers.

Let

and

Therefore,

x, xy, xy', xy', be the numbers.

x+xy=15,

xy3+xy3=60.

Taking the first value of x, and the corresponding value of y, we obtain the series

5, 10, 20, 40;

which numbers may be easily verified.

Taking the second value of x, and the corresponding value of y, we obtain the series

−15, +30, −60, +120;

which numbers also perfectly satisfy the problem understood algebraically. If, however, it is required that the terms of the progression be positive, the last value of x would be inapplicable to the problem, though satisfying the algebraic equation. Several of the following problems also have two solutions, if we admit negative values.

Prob. 2. There are three numbers in geometrical progression whose sum is 210, and the last exceeds the first by 90 What are the numbers?

Ans. 30, 60, and 120.

Prob. 3. There are three numbers in geometrical progression whose continued product is 64, and the sum of their cubes is 584. Required the numbers.

Ans. 2, 4, and 8.

Prob. 4. 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, and 27.

Prob. 5. Of four numbers in geometrical progression, the difference between the first and second is 4, and the difference between the third and fourth is 36. What are the numbers? Ans. 2, 6, 18, and 54.

Prob. 6. Of four numbers in geometrical progression, the

sum of the first and third is a, the sum of the second and fourth What are the numbers?

is b.

Ans.

a3 a2b ab❜ b3 a2+b2 a2+b2 a2+b2 a2+b2

HARMONICAL PROGRESSION.

(247.) A series of quantities is said to be in harmonical progression when, of any three consecutive terms, the first is to the third as the difference of the first and second is to the difference of the second and third.

Thus the numbers

60, 30, 20, 15, 12, 10,

are in harmonical progression; for

60 20: 60-30:30-20
30

15:30-20: 20-15

20: 12: 20-15: 15-12

15: 10 :: 15-12: 12–10.

So, also, the numbers

1, 1, 1, 1, 1, %,

1, &c.,

form an harmonical progression.

(248.) The reciprocals of a series of terms in harmonical progression form an arithmetical progression.

Thus, the reciprocals of 60, 30, 20, &c., are

80, 30, 70, 15, 12, 10,

which are respectively equal to

do, do, do, do, do, do,

being an arithmetical progression whose common difference is 'ਨ•

If six musical strings of equal weight and tension have their lengths in the ratio of the numbers

the second will sound the octave of the first; the third will sound the twelfth; the fourth will sound the double octave ; the fifth will sound the seventeenth; and the sixth will sound the nineteenth, and so on. Hence the origin of the term har

nonical or musical proportion.

Let a, b, c be three quantities in harmonical progression; then

That is, an harmonical mean between two quantities is equal

to twice their product divided by their sum.

SECTION XV.

GREATEST COMMON DIVISOR.-CONTINUED FRACTIONS.-PERMUTATIONS AND

COMBINATIONS.

(249.) The greatest common divisor of two or more quantities is the greatest factor which is common to each of the quantities.

THEOREM.

The greatest common divisor of two quantities is the same with the greatest common divisor of the least quantity, and their remainder after division.

To prove this principle, let the greatest of the two quantities be represented by A, and the least by B. Divide A by B; let the entire part of the quotient be represented by Q, and the remainder by R. Then, since the dividend must be equal to the product of the divisor by the quotient + the remainder, we shall have

A=QB+R.

Now every number which will divide B will divide QB; and every number which will divide R and QB will divide R+QB or A. That is, every number which is a common divisor of B and R is a common divisor of A and B.

Again, every number which will divide A and B will divide A and QB; it will also divide A-QB or R. That is, every number which is a common divisor of A and B is also a common divisor of B and R. Hence the greatest common divisor of A and B must be the same as the greatest common divisor of B and R.