RULE. Multiply the sum of the two extremes into the number of terms, and divide the product by 2. The quotient will be the sum of all the series. Or multiply the sum of the two extremes by half the number of terms. (1) How many strokes do the clocks at Venice (which go on to 24 o'clock) strike in the compass of a natural day? (2) How many strokes does the hammer of a clock strike in 12 hours? (3) The length of a garden is 94 feet: now if eggs be laid along the pavement a foot asunder, and be fetched up singly to a basket, removed one foot from the first, how much ground does he traverse that does it? (4) Suppose 100 stones were placed in a right line, a yard distant from one another, and the first stone were one yard from a basket; I demand how many miles he must travel that gathers them singly into the basket. (5) A butcher bought 100 sheep, and gave for the first sheep 1s. and for the last 91. 19s. I demand what he gave for the 100 sheep. PROPOSITION II. When the two extremes and number of terms are given, to find the common difference. RULE. The difference of the two extremes divided by the number of terms less a unity or 1, the quotient will be the common difference. EXAMPLES. (6) One had 20 children that differed alike in their ages: the youngest was 5 years old, the eldest 43. What was the difference of their ages, and the age of each ? (7) A running footman, for a wager, is to travel from London to a certain place northwards in 19 days, and to go but 6 miles the first day, increasing every day's journey by an equal excess, so that the last day's journey may be 60 miles. 1 demand each day's journey, and what distance the place he goes to is from London. (8) A debt is to be discharged at 10 different payments in Arithmetical Progression: the first payment is to be 57. and the last 30l. What is the whole debt, and what must each payment be? PROPOSITION III. When the two extremes and the common difference are given, to find the number of terms. RULE. Divide the difference of the two extremes by the common excess or difference; add unity or 1 to the quotient, and the sum will be the number of terms. EXAMPLES. (9) A man being asked how many children he had, answered, that his youngest child was 5 years old, and the eldest 43, and that he had increased one in his family every two years. How many children had he? (10) A person travelling from London northward, went 6 miles the first day, and increased every day's journey 3 miles, till at last he went 60 miles in one day. How many miles did he travel? PROPOSITION IV. When the last term, the common difference, and the number of terms are given, to find the first term. RULE. Multiply the number of terms, less unity or 1, by the common difference; the product subtracted from the last term leaves the first. EXAMPLES. (11) A man in 19 days went from London to a certain place in the country; every day's journey was greater than the preceding one by 3 miles: his last day's journey was 60 miles. What was the first? (12) A person takes out of his pocket, at 10 different times, so many different numbers of guineas, every one exceeding the former by two; the last was 23. was the first? PROPOSITION V. What When the number of terms, common difference, and the sum of all the terms are given, to find the first term. RULE. Divide the sum of all the series by the number of terms, and from that quotient subtract half the product of the common difference multiplied by the number of terms less one, gives the first term. (13) A EXAMPLES. person is to receive 2751. at 10 different payments, each payment to exceed the former by 51. he is willing to bestow the first payment on any one that can tell what it is. What must the arithmetician have for his pains? (14) Suppose it be 100 leagues between London and Edinburgh, two couriers set out from each place on the same road; that from London towards Edinburgh travelling every day two leagues more than the day before; that from Edinburgh to set off one day after the other, travelling every day three leagues more than the preceding one, and that they meet exactly half way, the first at the end of five days, and the other at the end of four: how many leagues did each travel per day? PROPOSITION VI. When the first term, number of terms, and the common difference are given, to find the last term. 11 RULE. Subtract the common difference from the product of the number of terms, multiplied by the common difference: the remainder, added to the first term, will give the last. EXAMPLES. (15) What is the last term of an arithmetical progression, beginning at 6, and continuing by the increase of 3 to 19 places? (16) What is the last term of an arithmetical progression, beginning at 1, and continuing by the increase of 2 to 100 places? PROPOSITION VII. The first term, common difference, and number of terms being given, to find the sum of all the series. RULE. From the product of the number of terms in the common difference, subtract the common difference; and to the remainder add the double of the first term: half the product of that sum multiplied by the number of terms, gives the sum of all the series. EXAMPLE. (17) A gentleman bargains with a bricklayer to sink him a well 30 yards deep, upon these terms, viz. to pay him three shillings for the first yard, five for the second, seven for the third, &c. raising two shillings for every yard. What will be due to the bricklayer for completing the same? PROPOSITION VIII. The first term, the number of terms, and sum of all the terms being given, to find the common difference. RULE. Divide the double sum of all the series by the number of terms, and from the quotient subtract double the first term; divide the remainder by the number of terms lessened by unity; the quotient will be the common difference. EXAMPLES. (18) A gentleman travelled 200 miles in eight days, and every day travelled equally farther than the preceding day; it is known that the first day he travelled six miles. How many miles did he travel each of the other days? (19) A person travelled from London to York, being 200 miles, in 9 days, and every day travelled equally farther than the preceding day; it is known that the first day he travelled 4 miles. How many miles did he travel each of the other days? PROPOSITION IX. When one person or thing moves with an equal, and another the same way, by a progressive motion, to find what time the first will be overtaken. RULE. To double the space gone each day by the pursued, add the common difference of the pursuer's day's journey; from that sum subtract double the space he travelled the first day, and divide the remainder by the common difference: the quotient will give the number of days in which the pursued will be overtaken by the pursuer. EXAMPLES. (20) A noted highwayman having committed a robbery, not suspecting a pursuit, fled northward at the rate of nine leagues a day; one of Sir John Fielding's men followed him in a progressive motion, only three leagues the first day, five the second, seven the third, and so on, increasing every day's journey two leagues. In how many days was the highwayman overtaken? (21) Y. Z. made the following bet for 1000 guineas, to be decided the Monday, Tuesday, and Wednesday, in |