3. It is required to find 3 geometrical means between 6 and 486.

By formula (D),

r = √486 = √81 = 3.

Therefore, the series is 6, 18, 54, 162, 486, Ans.

4. Find the sum of the series 6, 2,,.... to infinity.

We have given, a = 6, r=; hence, by formula (C'),

5. Find the exact value of the decimal .454545.... to infinity. This is a circulating decimal, and may be expressed thus:

In all such cases, the repetend, taken with its local value, will be the first term of a geometrical series, of which the ratio will or some power of. In the present example we have

be

1. Find the sum of 9 terms of the series 1, 2, 4, 8, . . . .

Ans. 511.

2. Find the 8th term of the progression 2, 6, 18, 54, ....

3. Find the sum of 10 terms of the series 1, §, 4,

Ans. 4374.

287,.. Ans. 174075

59049

4. Find two geometrical means between 24 and 192.

Ans. 48, 96.

5. Find geometrical means between 3 and 768.

Ans. 6, 12, 24, 48, 96, 192, 384.

6. Find the value of 1 + + + + . . . . to infinity.

Ans. 4.

7. Find the value of § + 1 + 3 + 2% + to infinity.

....

Ans. 4.

8. Find the value of 5 + § +8 +27+ to infinity.

....

Ans. 7.

9. Find the value of the decimal .323232 .... to infinity.

Ans. 3.

10. Find the value of the decimal .212121. . . . to infinity. Ans.

11. Find the value of − 1 + 1 − +.... to infinity.

16

Ans.

12. Find the value of ++.... to infinity.

15. The sum of a geometrical series is 1785, the ratio 2, and the number of terms 8; find the first term.

Ans. 7.

16. The sum of a geometrical series is 7812, the ratio 5, and the number of terms 6; find the last term. Ans. 6250.

17. The first term of a geometrical series is 5, the last term 1215, and the number of terms 6. What is the ratio? Ans. 3.

18. A man purchased a house with ten doors, giving $1 for the first door, $2 for the second, $4 for the third, and so on. What did the house cost him? Ans. $1023.

PROBLEMS IN GEOMETRICAL PROGRESSION

TO WHICH THE FORMULAS DO NOT

IMMEDIATELY APPLY.

365. The terms of a geometrical progression are represented in a general manner as follows:

x, xy, xy3, xy3,....

In the solution of problems, however, the following notation is generally preferable :

1st. When the number of terms is odd, the series may be represented thus:

2d. When the number of terms is even, the series may be expressed thus:

x2

x, y,

х

1. The sum of three numbers in geometrical progression is 26, and the sum of their squares 364. What are the numbers?

Let the numbers be denoted by x, √xy, y.

Then

and

x + √xy + y = 26 = a

=

x2 + xy + y2 = 364 b Transposing √xy in (1), squaring and reducing,

(1),

From (1) and (2),

x=2, and y= 18.

Hence, the numbers are, 2, 6, 18, Ans.

2. The sum of four numbers in geometrical progression is 15 or a, and the sum of their squares 85 or b. What are the numbers?

Taking the proper notation for an even number of terms, we

x2 + y2 = = s2 — 2p,

Assume x+y=s, and xy=p; then by 301,

Substituting the values of (x + y) and (x2 + y2), in (1) and (2),

x3 + y3 = 83 — 3sp.

Squaring (3), and then transposing 2xy, or 2p,

(3),

(4).

Clearing (3) of fractions, and putting xy=p in second member,

or,

Substituting this value of p in (6), and reducing,

Substituting the values of a and s in (7), we obtain

3. There are three numbers in geometrical progression; their sum is 21, and the sum of their squares is 189. Find the numbers. Ans. 3, 6, 12.

4. Divide the number 210 into three parts, so that the last shall exceed the first by 90, and the parts be in geometrical progression. Ans. 30, 60, and 120.

5. The sum of four numbers in geometrical progression 30; and the last term divided by the sum of the mean terms gives 1. What are the numbers? Ans. 2, 4, 8, and 16.

6. The sum of the first and third of four numbers in geometrical progression is 148, and the sum of the second and fourth is 888. What are the numbers? Ans. 4, 24, 144, and 864.

7. It is required to find three numbers in geometrical progression, such that their sum shall be 14, and the sum of their squares 84. Ans. 2, 4, and 8.

8. 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. What are the numbers? Ans. 1, 3, 9, and 27.

9. There are three numbers in geometrical progression; the sum of the first and second is 20, and the difference of the second and third is 30. What are the numbers? Ans. 5, 15, 45.

10. The continued product of three numbers in geometrical progression is 216, and the sum of the squares of the extremes is 328. What are the numbers ? Ans. 2, 6, 18.

11. The sum of three numbers in geometrical progression is 13, and the sum of the extremes being multiplied by the mean, the product is 30. What are the numbers? Ans. 1, 3, and 9.

12. There are three numbers in geometrical progression; their continued product is 64, and the sum of their cubes is 584. What are the numbers? Ans. 2, 4, 8.

13. There are three numbers in geometrical progression; their continued product is 1, and the difference of the first and second is to the difference of the second and third as 5 to 1. What are the numbers?

Ans. 5, 1, .