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root of which is 431; from this, subtracting 271⁄2, we get 16, for the number of terms.

2. The first term of an arithmetical progression is 2, the common difference is 3, and the sum of all the terms is 442. What is the number of terms? Ans. 17.

3. The first term of an arithmetical progression is, the cómmon difference is, and the sum of all the terms is 601. What is the number of terms?

CASE XVIII.

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

RULE.

To twice the last term add the common difference, divide the sum by twice the common difference, square the quotient, and from this square, subtract the quotient of twice the sum of the terms divided by the common difference; extract the square root of the remainder: then subtract this root from the quotient of the sum of twice the last term and common difference, divided by twice the common difference..

Examples.'

1. The common difference of the terms of an arithmetical progression is, the last term is 351, and the sum of all the terms is 1900. What is the number of terms?

In this example, twice the last term, increased by the common difference, is 71, which, divided by twice the common difference, gives 107; this, squared, becomes 11449. Again, twice the sum of all the terms, divided by the common difference, gives 11400; this, subtracted from 11449, gives 49, whose square root is 7. Subtracting this root from 107, we get 100, for the number of terms.

2. The common difference of the terms of an arithmetical progression is, the last term is 37, and the sum of all the terms is 601. What is the number of terms?

Ans. 26.

3. The common difference of the terms of an arithmetical progression is 1, the last term is 14, and the sum of all the terms 105. What is the number of terms?

CASE XIX.

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

RULE.

Divide the sum of the terms by the number of terms; from this quotient subtract half the product of the common difference into the number of terms, less one.

Examples.

1. The common difference of the terms of an arithmetical progression is 7, the number of terms 54, and the sum of all the terms is 10233. What is the first term?

In this example, the sum of the terms, divided by the number of terms, is 1891; half the product of the number of terms, less one, into the common difference is 1851, which, subtracted from 1891, leaves 4, for the first term.

2. The common difference of the terms of an arithmetical progression is, the number of terms is 59, and the sum of all the terms is 1180. What is the first term? Ans. 3.

3. The common difference of the terms of an arithmetical progression is 1, the number of terms is 30, and the sum of all the terms is 8321. What is the first term?

4. A father divides $2000 among five sons, so that each

elder should receive $40 more than his next younger brother. What is the share of the youngest?

CASE XX.

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

RULE.

From the square of the last term increased by half the common difference, subtract twice the product of the common difference into the sum of all the terms; extract the square root of the remainder, and to this root add half the common difference.

Examples.

1. The common difference of the terms of an arithmetical progression is 4, the last term is 1008, and the sum of all the terms is 127512. What is the first term?

In this example, the square of the last term, increased by half the common difference, is 1020100; twice the product of the common difference into the sum of all the terms is 1020096, which, subtracted from 1020100, gives 4, the square root of which is 2; this, increased by half the common difference becomes 4, for the first term.

2. The common difference of the terms of an arithmetical progression is 3, the last term is 49, and the sum of all the terms is 420. What is the first term?

Ans. 7.

3. The common difference of the terms of an arithmetical progression is 10, the last term is 1003, and the sum of all the terms is 50300. What is the first term?

CHAPTER VII.

GEOMETRICAL PROGRESSION.

63. A series of numbers which succeed each other regularly, by a constant multiplier, is called a geometrical progres

sion.

This constant factor, by which the successive terms are multiplied, is called the ratio.

When the ratio is greater than a unit, the series is called an ascending geometrical progression.

When the ratio is less than a unit, the series is called a descending geometrical progression.

Thus 1, 3, 9, 27, 81, &c., is an ascending geometrical progression, whose ratio is 3.

And 1,,,, &c., is a descending geometrical progression, whose ratio is

In geometrical progression, as in arithmetical progression, there are five things to be considered:

1. The first term.

2. The last term.

3. The common ratio.

4. The number of terms.

5. The sum of all the terms.

These quantities are so related to each other, that any three being given, the remaining two can be found.

Hence, there must be 20 distinct cases arising from the different combinations of these five quantities.

The solution of some of these cases requires a knowledge of higher principles of mathematics than can be detailed by arithmetic alone.

We will give a demonstration of the rules of some of the most important cases.

CASE I.

By the definition of a geometrical progression, it follows that the second term is equal to the first term, multiplied by the ratio; the third term is equal to the first term, multiplied by the second power of the ratio; the fourth term is equal to the first term, multiplied by the third power of the ratio; and so on, for the succeeding terms.

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

RULE.

Multiply the first term by the power of the ratio, whose exponent is one less than the number of terms,

Examples.

1. The first term of a geometrical progression is 1, the ratio is 2, and the number of terms is 7. term?

What is the last

In this example, the power of the ratio, whose exponent is one less than the number of terms, is 2=64, which, multiplied by the first term, 1, still remains 64, for the last term.

2. The first term of a geometrical progression is 5, the ratio is 4, and the number of terms 9. What is the last term? Ans. 327680.

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