EXAMPLES. 1. What is the duty on an invoice of silver and plated ware, imported from London, the cost exclusive of commissions, &c. being £.359 18 4, at 15 per cent. ad valorem ? 2. What will it amount to in a foreign vessel, at 161⁄2 per cent. ad valorem ? Ans. 290 dols. 4 cents. ...... The rates at which all foreign coins and currencies are estimated at the Custom-Houses of the United States. Each pound sterling of Great Britain, at Each livre tournois of France... Each florin or gilder of the United Netherlands Each mark banco of Hamburgh Each rix dollar of Denmark Each real of plate of Spain Each pagoda of India Each rupee of Bengal ... .... PROGRESSION Consists in two parts-ARITHMETICAL and GEOMETRICAL. ARITHMETICAL PROGRESSION Is when a rank of numbers increase or decrease regularly, by the continual adding or subtracting of some equal number: As 1, 2, 3, 4, 5, 6, are in Arithmetical Progression by the continual increasing or adding of one, and 11, 9, 7, 5, 3, 1, by the continual decrease or subtraction of two. NOTE. When any even number of terms differ by Arithmetical Progression, the sum of the two extremes will be equal to the two middle numbers, or any two means equally distant from the extremes: As 2, 4, 6, 8, 10, 12, where 6+ 8, the two middle numbers, are = 12 + 2, the two extremes, and 10+ 4 the two means 14. When the number of terms are odd, the double of the middle term will be equal to the two extremes, or of any two means As 1, 2, 3, 4, 5, equally distant from the middle term: where the double of 3 = 5 + 1 = 2 + 4 = 6. In Arithmetical Progression five things are to be observed, viz. 1. The first term. The sum of all the terms. 5. Any three of which being given, the other two may be found. The first, second and third terms given to find the fifth. RULE. Multiply the sum of the two extremes by half the number of terms, or multiply half the sum of the two extremes by the whole number of terms, the product is the total of all the terms. 1. EXAMPLES. How many strokes does the hammer of a clock strike in 12 hours? 12+1=13 en 13 x678 Ans. 2. A man buys 17 yards of cloth, and gave for the first yard 2s. and for the last 10s. what did the 17 yards amount to? Ans. £.5 28. 3. If 100 eggs were placed in a right line, exactly a yard asunder from one another, and the first a yard from a basket, what length of ground does that man go who gathers up these 100 eggs singly, returning with every egg to the basket to put it in ? Ans. 5 miles, 1300 yards. The first, second and third terms given to find the fourth. RULE. From the second subtract the first, the remainder divided by the third less one gives the fourth. EXAMPLES. 1. A man had 8 sons, the youngest was 4 years old, and the eldest 32, they increase in Arithmetical Progression ; what was the common difference of their ages? Ans. 4. 32-4-28 then 288-1-4 the common difference. 2. A man is to travel from Boston to a certain place in 12 days, and to go but 3 miles the first day, increasing every day by an equal excess, so that the last day's journey may be 58 miles; what is the daily increase, and how many miles distant is that place from Boston ? Ans. 5 miles daily increase. Therefore as 3 miles is the first day's journey; . 3+5 8 second ditto. 8+513 third ditto, &c. The whole distance is 566 miles. The first, second and fourth terms given to find the third. RULE. From the second subtract the first, the remainder divide by the fourth, and to the quotient add 1, gives the third. EXAMPLES. 1. A person travelling into the country, went 3 miles the first day, and increased every day by 5 miles, till at last he went 58 miles in one day; how many days did he travel? Ans. 12. 58-3-55 then 55÷5—11 and 11+1=12 the number of days. 2. A man being asked how many sons he had, said that the youngest was 4 years old, and the eldest 32, and that he increased one in his family every 4 years; how many had he? Ans. 8. The second, third and fourth given to find the first. RULE. Multiply the fourth by the third, made less by 1, the product subtracted from the second gives the first. EXAMPLES. 1. A man in 10 days went from Boston to a certain town in the country, every day's journey increasing the former by 4, and the last day he went was 46 miles; what was the first? Ans. 10 miles. 4 x 10-1 36 then 46-5610, the first day's journey. 2. A man takes out of his pocket at S several times, so many different numbers of shillings, every one exceeding the former by 6, the last 46; what was the first? Ans. 4. The second, third and fifth given to find the first. RULE. Divide the fifth by the third, and from the quotient subtract half the product of the fourth, multiplied by the third less 1, gives the first. EXAMPLE. A man is to receive £.360 at 12 several payments, each to exceed the former by £.4, and is willing to bestow the first payment on any one that can tell him what it is; what will that person have for his pains? Ans. £.8. 4 x 12 1 360 1230 then 30 8, the first payment. 2 The first, third and fourth given to find the second. RULE. Subtract the fourth from the product of the third, multiplied by the fourth, that remainder added to the first gives the second. EXAMPLE. What is the last number of an Arithmetical Progression, beginning at 6, and continuing by the increase of 8 to 20 places? Ans. 158. 20 × 8-8-152 then 152+6=158, the last number. ...... GEOMETRICAL PROGRESSION Is the increasing or decreasing of any rank of numbers by some common ratio, that is, by the continual multiplication or division of some equal number: As 2, 4, 8, 16, increase by the multiplier 2, and 16, 8, 4, 2, decrease by the divisor 2. NOTE. When any number of terms is continued in Geo ́metrical Progression, the product of the two extremes will be equal to any two means, equally distant from the extremes : As 2, 4, 8, 16, 32, 64, where 64 × 2=4 × 32=8×16=128. When the number of terms are odd, the middle term multiplied into itself will be equal to the two extremes, or any two means equally distant from the mean: As 2, 4, 8, 16, 32, where 2x-32=4×16=8×8=64. In Geometrical Progression the same five things are to be observed as in Arithmetical, viz. NOTE. As the last term in a long series of numbers, is very tedious to come at, by continual multiplication; therefore, for the readier finding it out, there is a series of numbers made use of in Arithmetical Proportion, called indices, beginning with an unit, whose common difference is one, whatever number of indices you make use of, set as many numbers (in such Geometrical Proportion as is given in the question) under them: As 1, 2, 3, 4, 5, 6 indices. 2, 4, 8, 16, 32, 64 numbers in Geometrical Proportion. But if the first term in Geometrical Proportion be different from the ratio, the indices must begin with a cypher. As 0, 1, 2, 3, 4, 5, 6 indices.. 1, 2, 4, 8, 16, 32, 64 numbers in Geometrical Proportion. |