GEOMETRICAL PROGRESSION.

By Geometrical Progression is meant, a series of terms, which increase by a uniform multiplier, or decrease by a constant divisor. The multiplier or divisor is the ratio.

If I hire a man for six months and engage to pay him 3 dollars for the first month, and double the sum for the second, and double the second for third, and so on, what must I pay him for the last?

I can ascertain this by multiplying the wages of the first month to find the wages of the second, and so on.

3 Wages of the first month.

2

In this example, there are given the first term, the number of terms, and the ratio, to find the last.

It will be easily seen that, in producing each term of the series, the ratio is as many times a factor less 1, as the number of terms. (The ratio has been 5 times used to obtain the wages of the sixth month, 4 times to obtain the wages of the fifth month, &c.) The first term must always be a factor. Any term of the series is the product of the ratio raised to a power, whose index is one less than the number of the term, multiplied by the first term. Hence we may obtain the desired answer, in a more expeditious mode than by the process above.

Two being the ratio, if raised to the 5th power, (1 less than the number of terms,) it becomes 32, which multiplied by the first or least term is 96, the answer. Hence the

RULE 1.

When the ratio, less extreme and number of terms are given, to find the last term or greatest extreme,

Raise the ratio to a power one less than the number of

terms, multiply that power by the least extreme, and the product is the answer.

RULE 2.

When the first term, last term and ratio are given, to find the sum of the series,

Multiply the last term by the ratio, and from the product subtract the first. Divide the remainder by the ratio less 1, and the quotient will be the answer.

Example. If the first term is 2, the last 4374, and the ratio 3, what is the sum of the series of numbers?

If the first and last terms (the extremes) and the ratio are given, to find the number of terms,

Divide the greatest term by the least, then find what power of the ratio will equal the quotient, and, to the index of that power add one, and this will be the number of terms.

Example. Least term 2)4374 Greatest term.

2187

3 involved to the 7th power is 2187. Then 7+1=8 the number of terms.

RULE 4.

When the extremes and number of terms are given, to find the ratio;-Divide the greatest by the least term, and extract that root of the quotient whose index is denoted by the number of terms less 1, and the root will be the common ratio.

SECTION XXVIII.

SINGLE POSITION.

This name is given to a rule, where the conditions of the question are to furnish the necessary data for obtaining the

answer.

A said he had spent and of his money, which amounted to 72 dollars. What sum had he at first?

Suppose 150. Then half = 75

Here it will be easily seen, that 125 must bear the same relation to the supposed number, that 72 does to the true number.

The only object of using a supposed number, is to find such others, as are necessary in order to perform the operation.

But as questions of a character similar to the one above, are more easily answered by other means, the rule of Single Position is not important. To show this, the example may be resolved by Fractions.

A had spent = and

of

=

his money,
ğin all.
Then 72 is of the

He said this amounted to 72 dollars. sum he had at first; of 72 must be 122. If this is one sixth, the sum must have been 6 times as large=863 Ans. The following is the general

RULE.

To solve a question by Single Position, suppose any number at pleasure, and pursue the course rendered necessary by the conditions of the question. Then the result of the supposition will bear the same relation to the supposed number, as the given number does to the answer.

DOUBLE POSITION.

This rule is more complex than the last, requiring two supposed numbers. Questions, where the ratio between the required and supposed number differs from that of the given number to the required one, are readily answered by Double

Position. The demonstration of the rule is difficult without a knowledge of Algebra. The following is the general

RULE.

Suppose any two convenient numbers, and proceed with them as the conditions of the question require, and write down the errors. Multiply the first supposed number by the last error, and the last supposed number by the first error. If both supposed numbers are either too large or too small, divide the difference of the products by the difference of the errors. But if one is too large and the other too small, divide the sum of the products by the sum of the errors, and the quotient will be the answer. *

Example. What is that number which, on being increased by its half, fourth, and 5 more, will be doubled?

10-5=5)100(20 Ans.

In this example both errors are too large.

Most questions of this kind may be answered by other modes, and it is not frequently necessary to resort to this rule.

SECTION XXIX.

ALLIGATION MEDIAL.

If a merchant mixes 5 lbs. of tea worth 50 cents a pound, 6 pounds worth 75, and 8 pounds worth 90, what is a pound of this mixture worth?

* The true answer cannot be obtained by this rule, when the first error does not bear the same proportion to the second, as the difference between the true and first supposed number does to the difference between the true and second supposed number.

It is plain that if he ascertains the price of the whole, and then divides that amount by the whole number of pounds, the value of one pound of the mixture will be obtained.

If it were required to find the price of 6 pounds of the mixture, this would evidently be 6 times the price of one pound. The price of any part is easily found.

RULE.

Find the value of all the quantities mixed together and then divide the whole price by the whole number of the quantities mixed, and this will be the price of a unit of that quantity.

ALLIGATION ALTERNATE.

If I wish to mix together gold at 16, 18, 20 and 23 carats fine, to form a mixture of 19 carats fine. What quantity of each must I take?

It is evident that if I take 3 pounds at 23, and 4 pounds at 16, there must be 133 carats in the whole; and this number divided by 7, the number of pounds taken, will give 19 carats as the required fineness. A similar course with the others would produce a similar result. Hence the general

RULE.

Write the numbers under each other in the order of their value, and at the left hand place the given number. Join the numbers, so as to have one greater joined to one less than the given number. Take the difference between the given number and each price, placing the result opposite to the number by which it is obtained, and against that with which it is joined. The quantity then standing against each given number will be the quantity or amount of that number to be taken. Example. 16 4 pounds at 16 carats fine.

18

66

18

66

19