2. Geometrical Progression.

A series of numbers is said to be in Geometrical Progression, when its terms increase by a constant multiplier, or decrease by a constant divisor. Thus, 2, 4, 8, 16, 32, &c. and 27, 9, 3, 1, are series in geometrical progression, the one increasing by a constant multiplication, by 2, and the other decreasing by a constant division, by 3. The number by which the series is constantly increased or diminished, is called the ratio.

Problem I.

The first term, the last term, and the ratio given to find the sum of the series.

RULE.-Multiply the last tern by the ratio, and from the product subtract the first term, and the remainder divided by the ratio, less 1, will give the sum of the series.

The reason of the rule may be shown thus: take any series, as 1, 3, 9, 27, 81, 243, &c. multiply it by the ratio, and it will produce the series, 3, 9, 27, 81, 243, 729, &c. Let the given series be what it will, it is plain that the sum of the second series will be as many times that of the first as is expressed by the ratio. Now subtract the first series from the second, and it gives 729-1, which is evidently as many times the sum of the first series as is expressed by the ratio less 1; consequently, 728 the sum of the proposed series, and is the rule; or 729 is the last term multiplied by the ratio, 1 is the first term, and 3-1 is the ratio less 1, and the same will hold, whatever be the series.

When a geometrical series consists of an even number of terms, the product of the extremes is equal to the product of any two means equally distant from the extremes; and when the number of terms is odd, the product of the extremes is equal to the square of the middle term, or to the product of any two means equally distant from them.

Problem II.

The first term and ratio given to find any other term assigned.

RULE.*-1. Write a few of the leading terms of the series, and place their indices over them, beginning with a cipher, and add together the most convenient indices to make an index less by 1 than the number, expressing the place of the term sought.

2. Multiply the terms of the series belonging to those indices together for a dividend, and raise the first term to a power whose index is 1 less than the number of terms multiplied for a divisor; divide the dividend by the divisor, and the quotient will be the term sought.

NOTE. When the first term of the series is equal to the ratio, the indices must begin with a unit, and the indices added must make the entire index of the term required; and the product of the several terms found as above, will be the term required.

When the first term is equal to the ratio, the reason of the rule is obvious; for as every term is some power of the ratio, and the indices point out the number of factors, it is evident from the nature of multiplication, that the product of any two terms will be another term corresponding with the index, which is the sum of the indices standing over those respective terms. And when the series does not begin with the ratio, it will be seen that every term after the two first, contains some power of the ratio, multiplied by the first term, and therefore the rule is in this case equally evident.

QUESTIONS.

4. What is the rule?

1. When is a series of numbers in Geometrical Progression?

2. What is meant by the ratio?

3. What is the first problem?

5. What is the second problem?
6. What is the rule?

3. Annuities.

An Annuity is a sum of money payable every year for a certain number of years, or forever.

When the debtor keeps the annuity in his own hands beyond the time of payment, it is said to be in arrears.

The sum of all the annuities for the time they have been forborn, together with the interest due upon each, is called the amount.

If an annuity be to be bought off, or paid all at once at the beginning of the first year, the price which ought to be given for it, is called the present worth.

Case I.

To find the amount of an annuity at simple interest.

RULE.* Find the sum of the natural series of numbers, 1, 2, 3, &c. to the number of years less 1; multiply this sum by one year's interest of the annuity, and the product will be the whole interest due upon the annuity; to this product add the product of the annuity and time, and the sum will be the amount required.

*Whatever the time is, there is due upon the first year's annuity, as many years' interest as the whole number of years less 1; and gradually one less upon every succeeding year to the last but one, upon which there is due only one year's interest, and none upon the last: therefore, in the whole, there is due as many year's interest of the annuity, as the sum of the series, 1, 2, 3, 4, &c. to the number of years less 1. Consequently, one year's interest, multiplied by this sum, must be the whole interest due; to which, if all the annuities be added, the sum is plainly the amount.

Case II.

To find the present worth of an annuity at simple interest.

RULE.*-Find the present worth of each year by itself, discounting from the time it becomes due, and the sum of all these will be the present worth required.

Position is the rule which discovers the true number by the use of false, or supposed numbers. It is of two kinds, single and double

SINGLE POSITION.

Single Position teaches to resolve those questions whose results are proportional to their suppositions.

RULE. Take any number and perform the same operations with it as are described to be performed in the question: Then say, as the result of the operation is to the given number, so is the supposed number to the true one required.

*The reason of this rule is manifest from the nature of discount.

Double Position teaches to resolve questions by making two suppositions of false numbers.

RULE.*-1. Take any two numbers and proceed with each according to the condition of the question, noting the errors.

2. Multiply the first supposed number by the last error, and the last supposed number by the first error; and, if the errors be alike, (that is, both too great, or both too small,) divide the difference of the products by the difference of the errors; but if unlike, divide the sum of the products by the sum of the errors, and the quotient will be the answer.

Examples.

1. There is a fish, whose head is 9 feet long, his tail is as long as his head and half the length of his body, and his body is as long as his head and tail; what is the whole length of the fish?

*This rule is founded on the supposition that the first error is to the second, as the difference between the true and first supposed is to the difference between the true and second supposed number; when that is not the case, the exact answer to the question cannot be found by this rule.