The arithmetical operations respecting numbers in Geometrical Progression, are commonly limited to the finding of the last or any assigned term in the series, and of the sum of all the terms in the series.

To find the last term of a Geometrical Progression.

Rule. Multiply the first term by such a power of the ratio as is denoted by the number of times less one.

EXAMPLE.

To find the 7 th term of a series, in which the first term is 5, and the ratio is 4.

first term 5 X 4096 = 20480 7 th term.

EXERCISES.

Ex. 1. What is the last term of the series having 9 terms, of which the first is 3, and the ratio 3?

Answer 19683.

Ex. 2. The first term of a geometrical series is 2, the number of terms 12, and the ratio 5, required the last term. Last Term 97656250.

To find the sum of a series in Geometrical Progression. Rule. Multiply the last term by the ratio, and divide the difference between this product and the first term, by the difference between 1 and the ratio, and the quotient will be the sum required.

To find the sum of a series, the first term being 5, the ratio 4, and the number of the terms 7.

Ex. 1. The first term of a geometrical series is 1, the last term is 65536, and the ratio is 4, what is the sum of the terms? Answer 87381.

Ex. 2. What would be the value of 12 books, the 1 st being rated at 1 d. the second 2 d. the third 4 d. and so on doubling the value for each book? Answer £ 17 1 3.

Ex. 3. What sum of money will be paid in a year by paying 1 farthing the 1 st month, 3 farthings the second, and so on tripling each payment to the end of the 12 months?

Answer £276 15 10

ALLIGATION.

Alligation is the method of determining the amount of the different quantities having different rates that will be required to produce a given rate.

It is usual to separate calculations of this nature into three cases, according to whether neither of the quantities is limited, whether one is limited, or whether the total is limited. The first is called Alligation Alternate, of which the second and third cases are considered as branches, and which are called Alligation Partial and Alligation Total.

Rule. Arrange the given rates, so that the mean rate may be between the higher and the lower rates.

Then taking the given rates in pairs, find the difference between the mean rate and each of the two rates, and apply the difference to the alternate rate.

N. B. If the number of the given rates is odd, one upper or one lower rate must be at least twice used with the opposite

rates.

EXAMPLE.

To find what number of Pounds at 7 s., 8 s., 10 s., and 12 s. will be required to form a mean value with 9 s. per lb.

So also 3 lb. at 7 s., 1 lb. at 8 s., 1 lb. at 10 s., and 2 lb. at 12 s., make a total of 7 lb. worth 63 s. or 9 s. per lb.

It is to be observed, that when any arrangement of this sort has been made, the different results may be combined and the amounts used; as in the above, 4 lb. of the 1 st at 7 s., 4 lb. at 8 s., 3 lb. at 10 s., and 3 lb. at 12 s., make a total of 14 lb., which amount to 126 s., and therefore average 9 s. per lb.

N. B. What is usually called Alligation Medial, is in business called the finding of an average rate or price; for which see page 94.

CASE 2. PARTIAL.

Rule. Find any set of quantities which will produce a mean rate equal to the given mean rate; then if this obtained quantity of the given rate is equal to the given quantity, the other quantities will also be equal to the quantities required.

But if the obtained quantity is not equal to the given quantity, divide the number of the given quantity by the number of the obtained quantity, and multiply each of the obtained quantities by the quotient.

Example. To find how many lb., at the rate of 5 s. and 12 s per lb., are sufficient with 10 lb. at 7 s. per lb., to make a mean value of 8 s. per lb.

Rule.

Total 30 lb. at 8 s. = 240

CASE 3. TOTAL.

Proceed as with Alligation Alternate, and if the total is not equal to the given amount, divide the number of the given total by the number of the obtained total, and multiply each obtained quantity by the quotient.

Example. To find how many ounces of Gold 18 carats, 20 carats, and 23 carats fine, will be required to make up 12 oz. of Gold 22 carats fine.

EXERCISES.

Ex. 1. What proportions of Silver 216 dwts. fine, and 231 dwts. fine, may be used to make a purity equal to standard, or 222 dwts. fine?

Answer 9 oz. 216 fine, and 6 oz. 231 fine.

Ex. 2. What proportions of Silver 210 dwts. fine, 217 dwts. fine, and 228 dwts. may be used to make a purity equal to standard?

Answer 6 oz. 210 fine, 6 oz. 217 fine, and 17 oz. 228 fine.

Ex. 3. What proportions of Gold 23 carats 1 gr. or 93 grs. fine, 22 carats 3 grs. or 91 grs. fine, and 18 carats or 72 grs. fine, are required to make the standard fineness of 22 carats or 88 grs. fine?

viz.

Answer 16 oz. 93 fine, 16 oz. 91 fine, and 8 oz. 72 fine.
Ex. 4. What proportions of Spirit of the following strengths,

22 per cent. over proof or 122 strong,

7 per cent. over proof or 107 strong, 12 per cent. under proof or 88 strong,

are required to be mixed with 66 gallons of Spirits 10 per cent. under proof, or 90 strong, to make up a strength equal to proof, or 100 strong.

Answer 30 gal. 22 per cent. o.; 36 gal. 7 per cent. o.; 21 gal. 12 per cent. u.; and 66 gal. 10 per cent. u.

viz.

Ex. 5. What proportions of Spirit of the following strengths,

are required to make up 100 gallons of Spirit 4 per cent. under proof?

gal. 4 per cent. o. ; gal. 16 per cent. u.

Answer 284 gal. 10 per cent. o.; 19 19 gal. 12 per cent. u.; and 33 Ex. 6. I have an order for 1200 pieces of printed Calicos, to average 11 d. per yard; what quantities of the following different rates can I supply to execute this; viz. 13 d. per yard, 11 d. per yard, 10 d. per yard, and 81 d. per yard.

Answer 500 ps. at 131 d., 150 ps. at 11 d., 100 ps. at 101 d., and 450 ps. at 83 d.

POSITION.

Position, Supposition, or the Rule of False, is the method of determining the true quantity from one or more assumed quantities. It is therefore divided into two parts, called Single and Double Position.

SINGLE POSITION.

Rule. Assume any quantity similar to that which is given, and work with it according to the conditions of the calculation. Then if the assumed result is not the same as the given result, say, as the assumed is to the true result, so is the assumed quantity to the true quantity.

EXAMPLE.

To find that number of which the half, quarter, and eighth

Ex. 1. Three numbers amount together to 120; the 1 st is once and one fourth of the second; and the second is once and one third of the third; what are they? Answer 50, 40, and 30.

Ex. 2. The price of a horse, chaise, and harness, is £ 100; the horse is worth twice the chaise, and the chaise is worth three times the harness; what is the value of each?

Answer £ 60, £ 30, and £ 10. Ex. 3. In travelling a journey of 250 miles, supposing I increase my pace each day, one third, one fourth, one fifth, and one sixth of the preceding day's journey; how far should I go each day? Answer 30, 40, 50, 60, and 70 miles.