Whitsun-week, on Barnham Downs, between the again for the next 10; B. takes the next, and so for XXXV. GEOMETRICAL PROGRESSION IS when any rank or series of numbers increase by one common multiplier, or decrease by one common divisor. As 2. 4. 8. 16. 32. 64. Here the common multiplier or ratio is 2. Also 729. 243. 81. 27. 9. 3. Here the common divisor or ratio is 3. In any series of numbers in geometrical progression, the product of the two extremes is equal to the product of any two means that are equally distant from the extremes. As 3. 9. 27. 81. 243. 729 Here 3 x 729-27 x 819 × 243-2187. When the number of terms are odd, the middle term multiplied into itself will be equal to the product of the two extremes, or any two means equally distant from the said mean or middle term. As 3. 6. 12. 24. 48. 12 x 12 6 × 24-48 × 3=144. In geometrical progression the same five things are to be observed as in arithmetical progression, viz. 1. The first term. 2. The last term. 3. The number of terms. 4. The ratio. 5. The sum of the terms. Any three of these being known, the rest may be found. If to any series of numbers in geometrical proportion, when the first term is not a unit, or the same as ratio, but not a unit, there be assigned a series of numbers in arithmetical progression, beginning with a unit or 1, and whose common difference is 1, called indices or exponents. 1. 2. 3. 4. 5. 6. 7. Indices. 2. 4. 8. 16. 32. 64. 128. Thus progression. Number in geometrical The addition or subtraction of the indices, or numbers in arithmetical progression, directly corresponds with the product or quotient of their respective terms or series in geometrical progression. But if the series begin with unity, or 1, the indices must begin with a cipher. Thus (0, 1, 2, 3, 4, 5, 6, 7, &c. Indices. 1, 2, 4, 8, 16, 32, 64, 128. Now by these indices, and a few of the first terms, the last term, or any distant one, may be speedily found, without producing the whole series. PROPOSITION I. When the first term is unity, the ratio and number of terms being known, to find the last or any remote term. RULE. Find a few of the leading terms, over which place their indices, as before directed; then find what figures of the indices, which added together will give the index of the term wanted multiply the number standing under such indices into each other, and the last product will be the term required. Note. When the indices begin with a cipher, the sum of the indices made choice of must be always one less than the number of terms given in question, as 1 in the indices stands over the second term. EXAMPLES. (1) A boy agrees, for 16 oranges, to pay only the price of the last, reckoning a farthing for the first, and a halfpenny for the second, &c. doubling the price to the last. How much did he give for them? (2) A man bought a horse, and by agreement was to give what the last nail would come to, at a farthing for the first nail, two for the second, four for the third, &c. There were 4 shoes, and 9 nails in each shoe. I demand the price of the horse. PROPOSITION II. In any series, not proceeding from unity, the ratio and first term being given, to find any remote term, without producing all the intermediate terms. RULE. Proceed as in the last proposition, only observe to divide every product by the first term, and the quotient will be the term required. EXAMPLES. (3) A person dying, left 11 children, to whom, and to his executor, he bequeathed in the manner following, viz. To his executor, for seeing his will performed, 10l.; the youngest child to have 307.; and so on, every child to exceed the next younger in triple proportion. What will be the share of the eldest? (4) A nobleman dying, left 10 sons, to whom he left a certain sum of money to be divided among them, viz. The youngest son to have 500l. the second to have as much and half as much, and so on, every one to exceed the next youngest in the same ratio of 1. What is the share of the eldest? PROPOSITION III. When the first term, ratio, and number of terms are given, to find the sum of all the terms. RULE. Find the last term as before, from which take the first; divide the remainder by the ratio, less one, and to that quotient add the last term; which gives the sum required. EXAMPLES. (5) On New-year's day a gentleman married, and received of his father-in-law a guinea, on condition that he was to have a present on the first day of every month for the first year, which should be double still to what he had the month before. What was the young lady's portion? (6) One, at a country fair, had a mind to a string of 20 fine horses: but not caring to take them at 20 guineas per head, the jockey consented that he should, if he thought good, pay but a single farthing for the first, doubling it only to the 19th, and he would give the 20th into the bargain. This being presently accepted, how were they sold per head? (7) A laceman, well versed in numbers, agreed with a gentleman to sell him 20 yards of rich gold brocaded lace, for 2 pins the first yard, 6 for the second, 18 for the third, and so on in triple proportion. I demand how much the lace produced. The pins afterwards sold at a farthing per 100. I demand whether the laceman gained or lost by the sale thereof, supposing the said lace to have been bought at 87. 1s. 8d. per yard. (8) A servant agreed with a master unskilled in numbers to serve him 11 years without any other reward for his service but the produce of a wheat corn for the first year, and that product to be sown the second year, and so on from year to year, until the end of the time, allowing the increase to be but tenfold proportion. I demand what the 11 years' service came to, supposing the sum of the whole produce to be sold at 4s. per bushel. Note.-7680 wheat corns, round and dry out of the middle of the ear, are computed to fill a statute pint. PROPOSITION IV. Of any decreasing series in whose last term is a cipher, to find the sum of those series. RULE. Divide the square of the first term by the difference between the said first term and the second term in the series; the quotient will be the sum of the series. EXAMPLES. (9) A great ship pursues a little one, steering the same way, at the distance of four leagues from it, and sails twice as fast as the small ship. It is asked how far the great ship must sail before it overtakes the lesser. (10) Suppose a ball to be put in motion by a force which drives it 12 miles the first hour, 10 the second, and so on, continually decreasing in proportion of 12 to 10, to infinity. What space would it move through? XXXVI. PERMUTATION, OR, VARIATION, IS the changing or varying the order of things, in respect of their places. RULE. Multiply all the given terms in a series of Arithmetical Progressionals continually, whose first term or common |