[The work in this question is so lengthy, we have been compelled to make use of two pages in our operation.}

4. What is the fifth root of 5?

5. What is the seventh root of 2?

11(1·2436, nearly, which is a trifle too great.

CHAPTER X.

ARITHMETICAL PROGRESSION.

81. A SERIES of numbers which succeed each other regularly by a common difference, is said to be in arithmetical progression.

When the terms are constantly increasing, the series is an arithmetical progression ascending.

When the terms are constantly decreasing, the series is an arithmetical progression descending.

Thus, 1, 3, 5, 7, 9, &c., is an ascending arithmetical progression; and 10, 8, 6, 4, 2, is a descending arithmetical progression.

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

1. The first term.

2. The last term.

3. The common difference.

4. The number of terms.

5. The sum of all the terms.

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

If we denote the five things by the numerals 1, 2, 3, 4, 5, they may be taken by threes, as follows:

1, 2, 3, giving 4 and 5, making 2 cases.

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

To give a demonstration to all the rules of these 20 cases, would be a very difficult task for the ordinary processes of arithmetic; we will, therefore, content ourselves with demonstrating a few of the most important of them.

Case I.

By our definition of an ascending arithmetical progression, it follows that the second term is equal to the first, increased by the common difference; the third is equal to the first, increased by twice the common difference; the fourth is equal to the first, increased by three times the common difference; and so on, for the succeeding terms.

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