variation in the rule. Hence, we may require the sum of any descending series, as 1, 1, 1, 1, &c., to infinity, provided we determine the LAST term. Now, we perceive the magnitude of the terms decrease as the series advances; the hundredth term would be extremely small, the thousandth term would be very much less, and the infinite term nothing; not too small to be noted, as some tell us, but absolutely nothing.

Hence, in any decreasing series, when the number of terms is conceived to be infinite, the last term, L, becomes 0, and equation (2) becomes

This gives the following rule for the sum of a decreasing infinite series:

RULE.-Divide the first term by the difference between unity

and the ratio.

EXAMPLES.

1. Find the value of 1, 2,, &c., to infinity.

a=1, r=2

Ans. 4.

2. Find the exact value of the series 2, 1, 1⁄2, &c., to infinity.

Ans. 4.

3. Find the exact value of the series 6, 4, &c., to infinity. Ans. 18.

4. Find the exact value of the decimal .3333, &c., to infinity. Ans.

This may be expressed thus: +, &c. Hence, a=,

5. Find the value of .323232, &c., to infinity. a=32, ar=1; therefore, r

Ans. 3.

6. Find the value of .777, &c., to infinity.

1 1 1

7. Find the sum of the infinite series 1++++,&c.

(ART. 114.) When three numbers are in geometrical progression, the product of the extremes is equal to the square of the mean.

This principle is obvious from the general series

a, ar, àr2, ar3, art, ar®, &c.

Taking any three consecutive terms anywhere along the series, we observe, that the product of the extremes is equal to the square of the mean.

That is, if the three terms taken, are a, ar, ar2,

a2r2=(ar)2

If ar2, ar3, ar1 are the three terms,

ar2 Xar1=(ar3)2

Hence, to find a geometrical mean between two numbers, we must multiply them together, and take the square root. If we take four consecutive terms, the product of the extremes will be equal to the product of the means.

(ART. 115.) This last property belongs equally to geometrical proportion, as well as to a geometrical series, and the learner must be careful not to confound proportion with a series.

a: ar::b: br, is a geometrical proportion, not a continued series. The ratio is the same in the two couplets, but the magnitudes, a and b, to which the ratio is applied, may be very different.

We may suppose a : ar two consecutive terms of one series, and b: br any two consecutive terms of another series having the same ratio as the first series, and, being brought together, they form a geometrical proportion. Hence, the equality of the ratio constitutes proportion.

EXAMPLES.

1. Find the geometrical mean between 2 and 8. Ans. 4. (Art. 114) √2X8=4

2. Find the geometrical mean between 3 and 12.

3. Find the geometrical mean between 5 and 80.

4. Find the geometrical mean between a and b.

Ans. 6.

Ans. 20.

Ans. (ab). 5. Find the geometrical mean between and 9. Ans. 3. 6. Find the geometrical mean between 3a and 27a.

Ans. 9a.

7. Find the geometrical mean between 1 and 9. Ans. 3.

8. Find the geometrical mean between

and 3. Ans. 6.

9. Find two geometrical means between 4 and 256.

N. B. When the two means are found, the series will consist of four terms, and 4 will be the first term and 256 will be the last term.

Comparing this with the general series,

Therefore, 16 and 64 are the means required.

10. Find three geometrical means between 1 and 16.

Here, the first term of the series is 1, the last term 16, and the number of terms 5, because three terms are required, and two are already given.

Now, by equation (1), L=arn-1

Therefore, the means required are 2, 4, and 8.

We may obtain the ratio when the first and last terms are given, by the following formula:

The first and last terms of a geometrical series are 2 and 162, and the number of terms 5; required the ratio.

Ans. 3.

The first term of a geometrical series is 28, the last term 18900, and the number of terms 5; what is the ratio ?

PROBLEMS

THAT

Ans. 5.

INVOLVE THE PRINCI

PLES OF GEOMETRICAL PROGRESSION.

(ART. 116.) When we wish to express three unknown quantities in geometrical progression, we may represent them by x, xy, y, or by x2, xy, y2, or by x, xy, xy2, for either of these correspond with Art. 114; that is, the product of the extremes is equal to the square of the mean.

When we wish to express four unknown quantities in geometrical progression, we may express them by x, xy, xy2, xy3, or by P, x, y, Q.

The object of this last notation, is to reduce P and Q to terms expressed by x and y, thus:

Taking the first three terms only, we shall have

Taking the last three terms only, we shall have

х

Therefore, four quantities in geometrical progression may be expressed by x and y only, and the terms stand symmetrically thus:

In a similar manner, we might express more terms by x and y only, and have them stand symmetrically, if it were proper to extend this subject in a work as elementary as this.

1. Three numbers are in geometrical progression, the sum of the first and second is 90, and the sum of the second and third is 180. What are the numbers?

Ans. 30, 60, and 120.

Represent the numbers by x, xy, and xy2.

2. The sum of three numbers in geometrical progression is 7, and the sum of their squares is 21. numbers?

What are the

Ans. 1, 2, 4.

Squaring (3), x2+2xy+y2=a2—2a√xy+xy

Subtracting (4) from (5), 2xy=a2-3a-2a/xy+2xy (6)

(5)