Hence, when we have given the sum of all the terms, the number of terms, and the ratio, to find the first term, we have this

RULE.

Multiply the sum of all the terms by the ratio, less one, divide the product by the power of the ratio, whose index is the number of terms, after diminishing it by one.

Examples.

1. The sum of all the terms of a geometrical progression is 262143, the number of terms is 9, and the ratio is 4.

is the first term?

What

In this example the sum of all the terms multiplied by the ratio, less one, is 262143×3=786429; the power of the ratio, whose exponent is the number of terms, is 49=262144, this, diminished by 1, becomes 262143; ... 786429, divided by 262143, gives 3, for the first term.

2. The sum of all the terms of a geometrical progression is 59174, the number of terms is 7, and the ratio is 7. What is the first term? Ans. 9.

3. If a debt of $4095 is discharged in 12 months, by pay. ing sums which are in geometrical progression, the ratio of which is 2, how much was the first payment?

CASE VII.

We have shown under case II., that the sum of all the terms, multiplied by the ratio, less one, is equal to the first term sub. tracted from the last term into the ratio; therefore the first term is equal to the product of the ratio into the last term, diminished by the product of the ratio, less one, into the sum of all the terms.

Hence, when we have given the sum of all the terms, the last term, and the ratio, to find the first term, we have this

RULE.

Multiply the last term by the ratio, and from the product subtract the product of the sum of all the terms into the ratio, less

one.

Examples.

1. The sum of all the terms of a geometrical progression is 436905, the last term is 327680, and the ratio is 4. What is the first term?

In this example we find the last term, multiplied by the ratio, to be 1310720. The product of the sum of the terms into the ratio, less one, is 1310715; .. 1310720-1310715-5, for the first term.

2. The sum of all the terms of a geometrical progression is 6139, the last term is 3072, and the ratio is 2. What is the first term? Ans. 6.

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

CASE VIII.

From the condition under case II., we see that the ratio, multiplied into the sum of all the terms diminished by the last term, is equal to the sum of all the terms, diminished by the first term.

Hence, when we have given the first term, the last term, and the sum of all the terms, to find the ratio, we have this

RULE.

Divide the sum of all the terms diminished by the first term, by the sum of all the terms diminished by the last term,

Examples.

1. The first term of a geometrical progression is 5, the last term is 327680, and the sum of all the terms is 436905. What is the ratio?

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 6128. 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?

NOTE. The demonstration of the rules for the 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. 3d. 3qrs.; 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?

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?

In this example the ratio raised to a power, whose expo

nent is the number of terms, less one, is 39=19683, this mul. tiplied 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, dimin ished by one, becomes 59048; .. 4648967136, divided by 59048, gives 78732, for the last term.

2. 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 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.

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.

What is

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