In this example, the sum of all the terms, diminished by the first term, is 436900, and the sum of all the terms, diminished by the last term, is 109225; .. 436900, divided by 109225, gives 4 for the ratio.

2. The first term of a geometrical progression is 6, the last term is 3072, and the sum of all the terms is 6138. What is the ratio? Ans. 2.

3. The first term of a geometrical progression is 7, the last term is 1240029, and the sum of all the terms is 1860040. What is the ratio?

Ans. 3.

NOTE. The demonstration of the rules for the four following cases have not been given; they may, however, be obtained by combining the conditions of some of the foregoing cases.

Case IX.

Given the first term, the ratio, and the sum of all the terms, to find the last term.

RULE.

To the first term add the product of the ratio, less one, into the sum of all the terms; divide this sum by

the ratio.

EXAMPLES.

1. The first term of a geometrical progression is 4, the ratio is 3, and the sum of all the terms is 118096. What is the last term?

In this example, the product of the ratio, less one, into the sum of all the terms is 236192, which, added to the first term, gives 236196; this, divided by the ratio, gives 78732, for the last term.

2. A man bought a certain number of yards of cloth,

giving 3 cents for the first yard, 6 cents for the second yard, 12 cents for the third yard, and so on, for the succeeding yards. If the whole number of yards cost $122.63, what did the last cost? Ans. $62.33. 3. A person bought a certain number of pears for £4 5s. 3 d. 3 far.; he gave 1 farthing for the first, 2 farthings for the second, 4 for the third, and so on, doubling each time. What did he pay for the last? Ans. £2 2 s. 8 d.

Case X.

Given the ratio, the number of terms, and the sum of all the terms, to find the last term.

RULE.

Raise the ratio to a power whose exponent is the number of terms, less one; multiply together this power, the sum of all the terms, and the ratio, less one; then divide this product by one less than the power of the ratio, whose exponent is the number of terms.

EXAMPLES.

1. The ratio of the terms of a geometrical progression is 3, the number of terms is 10, and the sum of all the terms is 118096. What is the last term?

19683;

In this example, the ratio, raised to a power whose exponent is the number of terms, less one, is 3o this, multiplied by the sum of all the terms, and the ratio, less one, is 19683 × 118096 × 2=4648967136; the power of the ratio, whose exponent is the number of terms, is 59049; this, diminished by 1, becomes 59048; .. 4648967136, divided by 59048, gives 78732, for the last term.

2 The ratio of the terms of a geometrical progres

sion is 3, the number of terms is 10, and the sum of all the terms is 295240. What is the last term?

Ans. 196830.

3. The ratio of the terms of a geometrical progression is 2, the number of terms is 11, and the sum of all the terms is 20470? What is the last term?

Case XI.

Ans. 10240.

Given the first term, the number of terms, and the last term, to find the sum of all the terms.

RULE.

Extract the root denoted by the number of terms, less one, of the last and first terms; then raise these roots to a power, whose exponent is the number of terms; then divide the difference of these powers by the difference of the roots.

EXAMPLES.

1. The first term of a geometrical progression is 1, the number of terms is 10, and the last term is 19683. What is the sum of all the terms?

In this example, we must extract the 9th root of the last and first terms, which give 3 and 1 for the roots; these must each be raised to the 10th power, which give 59049 and 1, the difference of which is 59048; this, divided by 3-1=2, gives 29524, for the sum of all the

terms.

2. The first term of a geometrical progression is 1, the last term is 2048, and the number of terms is 12. What is the sum of all the terms?

Ans. 4095.

3. The first term of a geometrical progression is 1, the last term is 10077696, and the number of terms is 10. What is the sum of all the terms?

Case XII.

Ans. 12093235.

Given the ratio, the number of terms, and the last term, to find the sum of all the terms.

RULE.

Raise the ratio to a power whose exponent is the number of terms; from this power subtract one, and multiply the remainder by the last term; divide this product by the product of the ratio, less one, into the power of the ratio, whose exponent is the number of terms, less one.

EXAMPLES.

1. The ratio of the terms of a geometrical progression is 2, the number of terms is 12, and the last term is 2048. What is the sum of all the terms?

In this example, the ratio, raised to a power whose exponent is the number of terms, is 212=4096; this, diminished by 1, becomes 4095, which, multiplied by 2048, becomes 8386560; again, the power of the ratio, whose exponent is one less than the number of terms, is 2048, which, multiplied by the ratio, less one, is not changed; .. 8386560, divided by 2048, gives 4095, for the sum of all the terms.

2. The ratio of the terms of a geometrical progression is, the number of terms is 8, and the last term is 106. What is the sum of all the terms?

Ans. 307.

3. The ratio of the terms of a geometrical progression is, the number of terms is 7, and the last term is 2582 073 What is the sum of all the terms?

4096

Ans. 591-741
4096

NOTE. The eight remaining cases in geometrical progression cannot be solved by the ordinary processes of arithmetic, but require for their solution a knowledge of logarithms, and algebraic equations above the second degree.

83. WHEN the ratio of a geometrical progression is less than a unit, the first term will be the largest, and the last term the least; the progression, will, in this case, be descending. But, if we consider the series of terms in a reverse order, that is, calling the last term the first, and the first the last, the progression may then be considered as ascending.

If a decreasing geometrical progression be continued to an infinite number of terms, we may neglect the last term as of no appreciable value; we can find the sum of such a progression by Case II., when it is modified, as follows:

Given the first term of a descending geometrical progression, and the ratio, to find the sum of all the terms, when continued to infinity.

RULE.

Divide the first term by a unit diminished by the ratio.

EXAMPLES.

1. What is the sum of all the terms of the infinite series 1,,,, &c.?