inch thick, is worth $100, what will be the dimensions of a similar plate, of the same metal, worth $800? 14. If a man can dig a cellar 12 feet long, 10 feet wide, and 4 feet deep, in 3 days, what will be the dimensions of a similar cellar, requiring 24 days to dig it, working at the same rate, and the ground being of the same degree of hardness? 15. If I put 2 tons of hay in a stack 10 feet high, how high must a similar stack be to contain 16 tons? 16. Four women bought a ball of yarn 6 inches in diameter, and agreed that each should take her share separately from the outer part of the ball: how much of the diameter did each wind off? ARITHMETICAL PROGRESSION. 392. An ARITHMETICAL PROGRESSION is a series of numbers in which each is derived from the one preceding, by the addition or subtraction of the same number. THE COMMON DIFFERENCE is the number which is added or *subtracted. 393. When the series is formed by the continued addition of the common difference, it is called an increasing series; and when it is formed by the subtraction of the common difference, it is called a decreasing series: thus, 2, 5, 8, 11, 14, 17, 20, 23, 23, 20, 17, 14, 11, 8, 5, 2, is an increasing series is a decreasing series. The several numbers are called terms of the progression The first and last terms are called the extremes, and the in termediate terms are called the means. 394. In every arithmetical progression there are five parts, any three of which being given or known, the remaining two can be determined. They are, - 395. Having given the first term, the common difference, and the number of terms, to find the last term. 1. The first term of an increasing progression is 4, the common difference 3, and the number of terms 10: what is the last term? 9 OPERATION. No. less 1 3 com. diff. 27 4 1st term. 31 last term. ANALYSIS.-By considering the manner in which the increasing progression is formed, we see that the 2d term is obtained by adding the common difference to the 1st term; the 3d, by adding the common difference to the 2d; the 4th, by adding the common difference to the 3d, and so on; the number of additions, in every case, being one less than the number of terms found. Instead of making the additions, we may multiply the common difference by the number of additions, that is, by 1 less than the number of terms, and add the first term to the product. Rule.-Multiply the common difference by 1 less than the number of terms: if the progression is increasing, add the product to the first term, and the sum will be the last term; if it is decreasing, subtract the product from the first term, and the difference will be the last term. Examples. 1. What is the 18th term of an arithmetical progression, of which the first term is 4, and the common difference 5? 2. A man is to receive a certain sum of money in 12 payments: the first payment is $300, and each succeeding pay ment is less than the previous one by $20: what will be the last payment? 3. What will $200 amount to in 15 years, at simple interest, the increase being $14 for the first year, $28 for the second, and so on? 4. Mr. Jones has 12 children. He gives, by will, $1000 to the youngest, $50 more to the next older, and so on to each next older $50: how much did the oldest receive? 5. A man has a piece of land 35 rods in length, which tapers to a point, and is found to increase rod in width, for every rod in length: what is the width of the wide end? 6. James and John have 100 marbles. It is agreed between them that John shall have them all, if he will place them in a straight line half a foot apart, and so that he shall be obliged to travel 300 feet to get and bring back the furthest marble; and also, if he will tell, without measuring, how far he must travel to bring back the nearest. How far? CASE II. 396. Knowing the two extremes of an arithmetical progression, and the number of terms, to find the common difference. 1. The two extremes of a progression are 4 and 68, and the number of terms 17: what is the common difference? OPERATION. 68 4 ANALYSIS.-Since the common difference multiplied by 1 less than the number of terms gives a product equal to the difference of the extremes, if we divide the dif ference of the extremes by 1 less than the number of terms, the quotient will be the common difference : hence, 17 - 1 = 16)64(4 Rule.-Subtract the less extreme from the greater, and divide the remainder by 1 less than the number of terms: the quotient will be the common difference. Examples. 1. A man started from Chicago and traveled 15 days; each day's journey was longer than that of the preceding day by the distance which he traveled the first day what was his daily increase if he traveled 75 miles the last day? 2. A merchant sold 14 yards of cloth, in pieces of 1 yard each; for the first yard he received $, and for the last $26: what was the difference in the price per yard? 3. A board is 17 feet long; it is 2 inches wide at one end, and 14 at the other: what is the average increase in width per foot in length? 4. The fourth term of a series is 12, and the eleventh is 33: find the intermediate terms. CASE III. 397. To find the sum of the terms of an arithmetical progression. 1. What is the sum of the series whose first term is 2, common difference 3, and the number of terms 8? Given series, 2 5 8 11 14 17 20 23 Same, order inverted, 23 20 17 14 11 8 5 2 Sum of both series, 25+25+25+25+25+25+25+25 ANALYSIS.-The two series are the same; hence, their sum is equal to twice the given series. But their sum is equal to the sum of the two extremes, 2 and 23, taken as many times as there are terms; and the given series is equal to half this sum, or to the sum of the extremes multiplied by half the number of terms. Rule. Add the extremes together, and multiply their sum by half the number of terms; the product will be the sum of all the terms. Examples. 1. What debt could be discharged in a year, by weekly payments in arithmetical progression, the first payment being $5, and the last $100? 2. A person agreed to build 56 rods of fence; for the first rod he was to receive 6 cents, for the second, 10 cents, and so on: what did he receive for the last rod, and how much for the whole ? 3. If a person travels 30 miles the first day, and a quarter of a mile less each succeeding day, how far will he travel in 30 days? 4. If 120 stones be laid in a straight line, each at a distance of a yard and a quarter from the one next to it, how far must a person travel who picks them up singly and places them in a heap, at the distance of 6 yards from the end of the line and in its continuation? CASE IV. 398. Having given the first and last terms, and the common difference, to find the number of terms. 1. The first term of an arithmetical progression is 5, the common difference 4, and the last term 41: what is the number of terms? OPERATION. 41536 9110 No. terms ANAJ YSIS. Since the last term is equal to the first term added to the product of the common difference, by one less than the number of terms (Art. 395), it follows that, if the first term be taken from the last term, the difference will be equal to the product of the common difference by 1 less than the number of terms: if this be divided by the common difference, the quotient will be 1 less thar the number of terms. |