metrical 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 sun 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 pecks did he raise in all, allowing the increase to have been in a fourfold ratio? Ans. 1398101 pecks.

139. When the ratio of a geometrical progression is less than a unit, the first term will be the largest, and the last term the least; the progression will, in this case, be descending; but if we consider the series of terms in a reverse order, that is, calling the last term the first, and the first the last, the progression may then be considered as ascending.

If a decreasing geometrical progression be continued to

an infinite number of terms, we may neglect the last term as of no appreciable value; we can find its sum by Case II., when it is modified, as follows:

Given the first term of a descending geometrical progression, and the ratio, to find the sum of all the terms, when continued to infinity.

RULE.

Divide the first term by a unit diminished by the ratio.

EXAMPLE.

1. What is the sum of all the terms of the infinite series 1, 4, 7, t, &c. ?

In this example, a unit, diminished by the ratio, is 1-, and the first term, 1, divided by, gives 2, for the sum of all the terms.

2. What is the sum of the infinite series 1, 3, 1, 27, &c. ? Ans. 14. 3. What is the sum of the infinite series, Tổõ, Too0) T21⁄4 ̄ ̄, &c. ? Ans. 4.

4. What is the sum of the infinite series, T†o, Tõ3õõ, Toooo, &c, ? Ans. 1.

ALLIGATION.

140. ALLIGATION is generally treated under two distinct heads, called Allegation Medial and Allegation AlterThe latter, however, belongs properly to the province of Algebra

nate.

ALLIGATION MEDIAL.

141. ALLIGATION MEDIAL teaches the method of finding the mean value of a compound, when its several ingredients and their respective values are given.

What is Alligation Medial?

Suppose a grocer mixes 140 pounds of tea, which is worth 8s. per pound; 200 pounds, worth 6s. per pound; and 160 pounds, worth 10s. per pound. What is a pound of the mixture worth?

140 pounds of tea, at 8s. per pound, is worth 140×8= 1120s.; 200 pounds, at 6s., is worth 200 ×6=1200s.; 160 pounds, at 10s., is worth 160 × 10=1600s. Therefore, the mixture, which is 500 pounds, is worth 1120+1200+ 1600=3920s. Hence, one pound of the mixture must be worth 322072}s.

Hence, to find the mean value of a compound, composed of several ingredients of different values, we have this

RULE.

Divide the sum of the values of all the ingredients by the sum of the ingredients.

Repeat this Rule.

EXAMPLES.

1. A wine-merchant mixed several sorts of wine, viz.: 32 gallons, at 40 cents per gallon; 15 gallons, at 60 cents per gallon; 45 gallons, at 48 gallons, at 85 cents per gallon. gallon of the mixture ?

27*

cents per gallon; and 8 What is the value of a

Therefore, one gallon of the mixture is worth $50-20

100

$0.502-50 cents and 2 mills.

2. A farmer mixed together 7 bushels of rye, worth 72 cents per bushel; 15 bùshels of corn, worth 60 cents per bushel; and 12 bushels of wheat, worth $1.20 per bushel. What is the value of a bushel of the mixture?

Ans. $0.8314.

3. A goldsmith melts together 11 ounces of gold 23 carats fine, 8 ounces 21 carats fine, 10 ounces of pure gold, and 2 pounds of alloy. How many carats fine is the Ans. 125.

mixture?

It will be understood that a carat is a 24th part. Thus, 21 carats fine is the same as 2 pure metal; in the same way, 23 carats fine is 22 pure metal.

4. On a certain day, the mercury in the thermometer was observed to stand 2 hours at 62 degrees, 4 hours at 70 degrees, 5 hours at 72 degrees, 3 hours at 59 degrees, and 1 hour at 75 degrees. What was the mean temperature for the fifteen hours? Ans. 67 degrees.

5. Suppose a ship sail at the rate of 5 knots for 3 hours, at 7 knots for 5 hours, and 8 knots for 4 hours. What is her rate of sailing during the 12 hours?

Ans. 65 knots.

6. A grocer mixes 30 pounds of sugar worth 10 cents per pound; 40 pounds worth 10 cents per pound; 24 pounds worth 11 cents per pound; and 60 pounds worth 13 cents per pound. What is a pound of the mixture worth? Ans. 115 cents.

ALLIGATION ALTERNATE.

142. ALLIGATION ALTERNATE is the reverse of Alligation Medial; that is, it teaches the method of finding the ingredients when their rates are given, so that the compound shall have a given value.

What is Alligation Alternate?

Suppose we wish to mix teas, which are worth 4 and 6 shillings per pound, so that the mixture may be worth 5 shillings per pound: it is obvious that we must take equal quantities of each; since the price of the one is as much less than the mean price, as the other is greater.

Again, suppose we wish to mix teas, which are worth 4 and 7 shillings per pound, so that the mixture may be worth 5 shillings. In this case the 7 shilling tea is 2 shillings above the average price, whilst the 4 shilling tea is but 1 shilling below: it will be necessary to use twice as much of the 4 shilling tea as of the 7 shilling tea; and in all cases it is obvious that the quantities to be used will be in the inverse ratio to the differences between their prices and the mean price.

When there are more than two simples they may be compared together in couplets, one term of which must obviously exceed the average price, while the other must be less.

CASE L

The rates of the several ingredients being given, to make a compound of a fixed rate.