CHAPTER XI.

GEOMETRICAL PROGRESSION.

82. A SERIES of numbers which succeed each other regularly, by a constant multiplier, is called a geometrical progression.

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 1.

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, as in arithmetical progression, it may be shown that 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. What is the last term?

In this example, the power of the ratio, whose exponent is one less than the number of terms, is 26=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.

3. A person traveling, goes 5 miles the first day, 10 miles the second day, 20 miles the third day, and so on, increasing in geometrical progression. If he continue to travel in this way for 7 days, how far will he go the last day? Ans. 320 miles.

Case II.

If we multiply all the terms of a geometrical progression by the ratio, we shall obtain a new progression, whose first term equals the second term of the old progression; the second term of our new progression will equal the third term of the old progression, and so on for the succeeding terms. Hence, the sum of the old progression, omitting the first term, equals the sum of the new progression, omitting its last term.

The sum

of the new progression is equal to the old progression repeated as many times as there are units in the ratio. Therefore, the difference between the new progression and the old progression is equal to the old progression repeated as many times as there are units in the ratio, less one. But we also know that the difference between these progressions is equal to the last term of the new progression diminished by the first term of the old progression; and, since the new progression was formed by multiplying the respective terms of the old progression by the ratio, it follows that the last term of the new progression is equal to the last term of the old progression repeated as many times as there are units in the ratio.

Therefore, the last term of the new progression, diminished by the first term of the old progression, is equal to the last term of the old progression repeated as many times as there are units in the ratio and diminished by the first term of the old progression. Hence, we finally

obtain this condition:

That the sum of all the terms of a geometrical progression, repeated as many times as there are units in the ratio, less one, is equal to the last term multiplied by the ratio and diminished by the first term.

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

RULE.

Subtract the first term from the product of the last term into the ratio; divide the remainder by the ratio, less one.

EXAMPLES.

1. The first term of a geometrical progression is 4, the last term is 78732, and the ratio is 3. What is the sum of all the terms?

In this example, the first term, subtracted from the product of the last term into the ratio, is 236192, which, divided by the ratio, less one, gives 118096, for the sum of all the terms.

2. The first term of a geometrical progression is 5, the last term is 327680, and the ratio is 4. What is the sum of all the terms? Ans. 436905. 3. A person sowed a peck of wheat, and used the

whole crop for seed the following year; the produce of this second year again for seed the third year, and so on. If, in the last year, his crop is 1048576 pecks, how many ecks did he raise in all, allowing the increase to have en in a four-fold ratio? Ans. 1398101 pecks.

Case III.

Since by Case I. the last term is equal to the first erm multiplied into a power of the ratio whose expoent is equal to the number of terms, less one, it follows hat the first term is equal to the last term divided by the power of the ratio whose exponent is one less than the number of terms.

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

RULE.

Divide the last term by a power of the ratio whose exponent is one less than the number of terms.

EXAMPLES.

1. The last term of a geometrical progression is 1048576, the ratio is 4, and the number of terms is 11. What is the first term?

10

=

In this example, the ratio, 4, raised to a power whose index is 10, one less than the number of terms, is 4 1048576; .. 1048576, divided by 1048576, gives 1 for the first term.

2. A man has 6 sons, among whom he divides his estate in a geometrical progression, whose ratio is 2;