152 ARITHMETICAL PROGRESSION. ARITHMETICAL PROGRESSION. Any rank or series of numbers, more than two, increasing or decreasing by uniforin difference, are said to be in Arithmetical Progression. When the numbers are formed by a continual addition of the common difference, they form an ascending series; but when they are formed by a continual subtraction of the common difference, they form a descending series. Thus, $ 3, 5, 7, 9, 11, 13, 15, &c. is an ascending series. 15, 13, 11, 9, 7, 5, 3, &c. is a descending series. The numbers which form the series are called the terms of the series. The first and last terms are called the extremes, and the other terms the means. There are five denominations in arithmetical progression, any three of which being given, the other two may be found, viz. 1st. The first term. The last term. When the first term, the number of terms, and the common difference, are given, to find the last term, Multiply the number of terms, less 1, by the common difference. and add the first term to the product for the last term. 1. A man bought 100 yards of cloth, giving 4 cents for the first yard, 7 for the second, 10 for the third, and so on, with a common difference of 3 cents; what was the cost of the last yard ? Ans. 301 cents=$3.01. 2. There are, in a certain triangular field, 41 rows of corn; the first row, in one corner, is a single hill, the second contains 3 hills, and so on, with a common difference of 2; what is the number of hills in the last row ? Ans. 81 hills. 3. A man on a journey travelled 20 miles the first day, 24 the second, and so on, increasing 4 miles every day-how far did he travel the twelfth day? Ans. 64 miles. 4. There is an arithmetical series consisting of 18 terms; the first term is 4, the common difference 12; what is the largest term ? Ans. 208. 5. If the first term be 3, the common difference 2, and the number of terms 19; what is the last term ? Ans. 39. Questions.-What is Arithmetical Progression ? The numbers that form the series, what are they called? What are the respective names of the five denominations in arithmetical progression; When the first term, common difference, and number of terms given, how is the last term found? When is the series ascending? When descending ? Ans. 5 years. ARITHMETICAL PROGRESSION. 153 When the extremes and number of terms are given, to find the common difference. Divide the difference of the extremes, by the number of terms, less 1, and the quotient will be the common difference. 1. If the extremes be 5 and 605, and the number of terms 151; what is the common difference? Ans. 4. 2. A man had 8 sons, whose ages differed alike; the youngest was 10 years old, and the eldest 45; what was the common difference of their ages ? Question.-How is the common difference found when the extremes and terms are given ? When the extremes and number of terms are given, to find the sum of all the terms. Multiply half the sum of the extreme, by the number of terms, and the quotient will be the answer, (or &c.) 1. If the extremes, be 5 and 605, and the number of terms 151, what is the sum of the series? Ans. 46055. 2. How many times does a common clock strike in 12 hours ! Ans. 78. 3. How many strokes do the clocks of Venice strike, which go to 24 o'clock, in the compass of one day? Ans. 300, Questions.-Having the first and last terms, and the number of terms given, how is the sum of all the terms found? How can the rule be inverted so as to admit of three different methods of solutions, to each question ? The last term, the number of terms, and the common difference being given, to find the first term. Multiply the number of terms, less i, by the common difference, which product subtract from the låst term, and the remainder is the first term. 1. If the common difference be 2, the number of terms 19, and the last term 39, what is the first term ? Ans. 3. 2. A man takes out of his pocket, at 8 several times, so many several numbers of dollars, every one exceeding the former by 6; the last was 54, what was the first? Ans. $12. Question.--Having the last term, the number of terms, and the common difference given, how is the first term found ? Given the extremes (the first and last terms,) and the common difference, to find the number of terms. Divide the difference of the extremes, by the common difference, and the quotient increased by 1 will be the number of terms required. 1. The extremes are 2 and 53, and the common difference 3; what is the number of terms? Ans. 18 terms. 2. A man going a journey, travelled the first day 7 miles, the last day 51 miles, and each day increased his journey by 4 miles ; how many days did he travel, and how far? Ans. 12 days, and 348 miles. Questions.--Having the extremes and common difference given, how is the number of terms found ? Having 7 and 51 extremes given, and having obtained (12) the extremes, how is the sum of the terms 348 found ? 154 GEOMETRICAL PROGRESSION. GEOMETRICAL PROGRESSION. A GEOMETRICAL PROGRESSION is a ratio or series of numbers increasing by a constant multiplier, or decreasing by a constant divisor. Thus, 1, 2, 4, 8, 16, 32, &c. is an increasing geometrical ratio. And, 16, 8, 4, 2, 1, .5, &c. is a decreasing geometrical ratio. The ratio is the multiplier or divisor, by which the series is founded. In Geometrical Progression there are five denominations, any three of which being given, the other two may be found. 1st. The first term. To raise a power or series of numbers by the ratio, we place the ratio at the left hand for the first power; this (first power) multiplied by the ratio (its square) gives the second power, the second by the ratio gives the third, and so on, until the power is 1 number less than the first term. GENERAL RULE. Raise the ratio to a power, one number less than the first term; and this power multiplied by the first term, the product will be the last term. If the first term be subtracted from the product of the last term and ratio, and the remainder divided by the ratio less 1, it will give the sum of the series. Or, 1. Raise the ratio to a power equal to the number of terms; 2. Subtract one from that power; 3. Multiply the remainder by the first term; 4. Divide this product by the ratio less one, the quotient will be the sum of the geometrical series. Ex.-1. A man bought 12 yards of cloth, giving 4 cents for the first yard, 12 cents for the second, and so on, in a threefold ratio: what did he pay for the last yard, and what was the amount? Thus, 3x9x27x81 x 243 x 729 243 729 2187 6561 2916 1458 1458 5103 177147 the 11th power 531441 4 first term. -1 Ist ans. $7085.88 last term, or yd. 531440 X3 ratio. X4 21257.64 3—1=2)2125760 -4 first term. $10628.80 Ans. 3-1=2921257.60 Ans. $10628.60 sum of the series. or, 729 2. What debt can be discharged in a year, by paying 1 cent the first month, 10 cents the second, and so on, each month in a tenfold proportion? $1111111111.11. 3. A man bought a horse, and by agreement was to give 1 cent for the first nail, 2 for the second, 4 for the third, &c.; there were 4 shoes, and 8 nails in each shoe: I demand what the horse was worth at that rate? Ans. $42949672.95. 4. A gentleman whose daughter was married on a new-year's day, gave her a dollar, promising to triple on the first day of each month in the year; to how much did her portion amount ? Ans. $265720. 5. A gentleman dying, left his estate to his five sons; to the youngest $1000, to the second $1500, and ordered, that each son should exceed the younger by the ratio of 1.5; what was the amount of the estate? Ans. $13187.50. Questions. What is Geometrical Progression? The ratio is what? How many denominations does the rule embrace? What are their respective names ? How do you raise the ratio to any given power ? Recite the general rule. UNITED STATES' DUTIES, DUTIES are imposed by law on goods, wares, and merchandise, imported into the United States, at certain rates per centum ad valorem. When a duty is said to be ad valorem, it is meant that it is at a certain rate on the whole value of the goods. The term is used to distinguish this class of duties from those imposed on the quantity; as a duty upon the gallon, pound, barrel, cwt., ton, &c. A written account or catalogue of articles sent to a purchaser or factor, with the prices and charges annexed, is called an invoice. The ad valorem rates of duty upon goods, wares, and merchandise, are estimated by adding 20 per cent to the actual cost thereof, if imported from the Cape of Good Hope, or from any other place beyond it; and 10 per cent. if imported from any other place or country, including all charges;, commissions, outside packages, and insurance excepted. CASE 1. To find the duty on any amount of goods, wares, or merchandise, at any given rate per centum ad valorem. RULE. Reduce the cost of the goods to Federal money, and add 20 per cent. if imported from or beyond the Cape of Good Hope, or 10 per cent. if imported front any other place. Then multiply the amount by the given rate per cent. and divide by 100, or remove the decimal point two figures to the left for the duty required. 1. What will be the duty on an invoice of woollen goods, EXAMPLES. 156 UNITED STATES' DUTIES. imported from London, which cost £1250 10s. sterling, at 30 per cent. ad valorem? £1250.5 sterling cost 4.44=£1 sterling 50020 50020 50020 10)5552.22,0 actual cost in federal money 555 22 2 ten per cent. added. 6107.442 30 Ans. $1832.23,260 duty required 2. What will be the duty on an invoice of silk goods, imported from France, which cost 2650 francs, at 20 per cent. ad valorem? Ans. $109.20,3+ 3. What will be the duty on an invoice of silk and cotton goods, imported from India, which cost 2500 rupees, at 25 per cent. ad valorem? Ans. $375. 4. What will be the duty on an invoice of raisins, imported from Spain, which cost 640 piasters 4 reals 24 mar. vadies=51241; reals plate, at 40 per cent. ad valorem? Ans. $225.487 17. CASE 2. To find the duty on any amount of goods, wares, or merchandise, at any given rate per pound, gallon, &c. RULE. Multiply the given rate per pound, gallon, &c. by the number of pounds, gallons, &c. and the product will be the duty required. EXAMPLES. 1. What will be the duty on 150 chests of Hyson tea, imported direct from China, in a vessel of the United States, weighing gross 11250lbs., tare 20lbs. per chest, at 40 cents per pound? Ans. $3300. 2. What will be the duty on 20 pipes of French brandy, fourth proof, containing 2520 gallons, at 48 cents per gallon? Ans. $1209.60. 3. What will be the duty on 25 hogsheads of brown sugar, 1 |