278, 279. ARITHMETICAL PROGRESSION. 119 2. If the first term of a se- 3. A man has 1:2 sons whose ries be 8, the last 108, and ages are in arithmetical prothe number of terms 21, what gression ; tha youngest is 2 is the common difference? years old, and the oldest 35; 1088:21–1–5 Ans. what is the common difference in their ages? Ans. 3 yrs. 278. If I give 3 cents for the first lemon, and 11 cents for the last, and the common difference in the prices be 2 cents, how many did I buy?. The difference of the extremes divided by the number of terms, less 1, gives the common difference (277); consequently the difference of the exiremes divided by the common difference, inust give the number of term. less 1 (11-3=8, and 8-21, and 4+1=) 5 Aus. Hence, III. The first term, the last term, and the common difference given to find the number of terms. RULE.--Divide the difference of the extremes by the common difference, and the quotient, increased by 1, will be the answer. 2. If the first term of a se- 3. A man on a journey travries be 8, the last 108, and the elled the first day 5 miles, the common difference 5, what is last day 35 miles, and increasthe number of terms ? ed his travel each day by 3 108–8:5=20, and 20+1= miles; how many days did ho 21 Ans. travel? Ans. 11, 279. If I buy 5 lemons, whose prices are in arithmetical progression, giving for the first 3 cents, and for the last 11 cents, whal do I give for he whole ? The mean, or average price of the lemons will obviously be half way between 3 and 11 cents the difference between 3 and 11 added 10 3 19 (11—3:25). 7, and 7, the mean price, multiplied by 5, ibe number of lemons, equals (785) 35 cents, the answer. Therefore, IV. The first and last term, and the number of terms given to find the sum of the series. RULE.-Multiply half the sum of the extremes by the number of terms, and the product will be the sum of the series. 2. How many times does a 3. Thirteen persons gave common clock strike in 12 presents to a poor man hours? arithmetical progression; the 1+12; 2X12—78 Ans. first gave 2 cents, the last 26 cents; what did they all give ? Ang $1.82 2. Geometrical Progression. 280. A Geometrical Progression is a series of terms which increase by a constant inultiplier, or decrease by a constant divisor, as 2, 4, 8, 16; 32, &c., increasing by the constant multiplier, 2, or 27, 9, 3, 1, 3, &c., decreasing by the constant divisor, 3. The multiplier or divisor, by which the series is produced, is called the ratio. 281. A person bought 6 brooms, giving 3 cents for the first; 6 cents for the second, 12 for the third, and so on, doubling the price to the sixth ; what was the price of the sixth? or, in other words, if the first term of a series be 3, the number of terms 6, and the ratio 2, what is the last term ? The first term is 3, the second, 3X2=6, the third, 6X%(3X2X2=) 12, the fourth, 12x2(3x2x2x2) 24, the fifth, 24*2=(3x2x2x2x2 ) 48, and the sixth, 48 X2=(3x2x2x2x2x2=) 96. Then 96 cents is ihe cost of the sixth broom. By examining the above, it will be seen, that the ratio is, in the production of each term of the series, as many times a factor, less one, as the number of terms, and that the first term is always employed once as a factor, or, in other words, any term of a geometrical series is the product of the ratio, raised to a power whose index is one less than the number of the term, multiplied by the first term. Note. If the second power of a number, as 22, be multiplied by the third power, 23, the product is 25. Thus, 223 X2-4, and 23=2x2x2=8, and 8X4-322x2x2x2x2; and, generally, the power produced by multiplying one power by another is denoted by the sum of the indices of the given powers. Hence, in finding the higher powers of numbers, we may abridge the operation, by employing as factors several of the lower powers, whose indices added together will make the index of the required power. To find the seventh power of 2, wc may multiply the third and fourth powers together, thus : 27-23X248x16=128. Ans. 1. The first term and ratio given to find any other terms RULE.--Find the power of the ratio, whose index is one less than the number of the required term; and multiply this power by the first term, the product will be the answer, if the series is increasing ; but if it is decreasing, divide the first term by the power. 1. The first term of a ge 2. The first term of a des ometrical series is 5, the creasing series is 1000, the ratiu 3; what is the tenth ratio 4, and the number of terni ? terms 5; what is the least 39334X35=81 X248 term ? Ans. 37%. 19683, and 19683 x 598415 Ans. 282. A person hought 6 brooms, giving 3 cents for the first, and 96 cents for the last, and the prices form a geometrical series, the ratio of which was 8; what was the cost of all the brooms ? The price would be the sum of the following series : 3+6+12+24+ 49+96=189 cents, Ans. If the foregoing series be multiplied by the ratio 2, the product is 6+12+24+487-96=192, whose sum is twice that of the first. Now, subtracting the first series from this, the remainder is 192 189=the sum of the first series. Had the ratió been other than 2, the remainder would have been as many times the sum of the series as the ratio, less 1, and the remainder is always the difference between the first terma and the product of the last term by the ratio. Hence, II. The first and last term, and ratio given to find the sum of the series. ĶULE.-Multiply the last term by the ratio, and from the product subtract the first term, the remainder divided by the ratio, less 1, will give the sum of the series. 2. The first term of a geo- second, $4 the third, and so on, metrical series is 4, the last each succeeding payment beterm 972, and the ratio 3; | ing double the last; and what: what is the sum of the series | wül be the last payment? $.$4095 the debt. 3—11972X3_451456 Aps. Ans. $2048 last pay't. NOTE.-'The marks drawn over the numbers show, that 4 must be taken from the product of 972, by 3, 5. A gentleman, being askand the remainder divided by Ziled to dispose of a horse, said =)2 2. This mark is called a vincu- he would sell him on condition him. of having 1 cent for the first 3. The extremes of a geo- | nail in his shoes, 2 cents for metrical progression are 1024 | the second, 4 cents for the and 59049, and the ratio 1} ; third, and so on, doubling the what is the sum of the series ? k price of every nail to 32, the Ans. 175099 number of nails in his four 4. What debt will be dis- | shoes; what was the price of charged in 12 months, by pay. the horse at that rate ? ing $1 the first month, $2 the Ans. $42949672.98. years, wbat 283. If a pension of 100 dollars per annum be forborne is there due at the end of that time, allowing compound interest at 6 per cent. ? Whatever the time, it is obvious that the last year's pension will draw po interest; it is, therefore, only $100; the last but one will draw interest one year, amounting to $106; the last but two, interest (compound) for 2 years, amounting to $112,36; and so on, forming a geometrical progression, whose first term is 160, the ratio 1.06, and the sum of this series will be the amount due. To find the last term (281) say, 1.065X 100133.82255776, the sixth term; and to find the sum of the series (282) say, 193.82255777X1.06-100=41.8519112256, which, dividod by 1.06—10.06 gives $697.5318576 Ans. or rum due. 984. A sum of money payable every year, for a number of year; called annuity. When the payment of an annuity is forborne, it is said to be in arrears. 1. What is the amount of an an- 2. If a yearly rent of 850 be for nuity of $40, to continue 5 years, borne 7 years, to what does it amount, allowing 5 per cent. compound inter- at 4 per cont. compound interest ? Ains. $221.025. Ans. $394.9L. 3. Duodecimals. 285. Of the various subdivisions of a foot, the following is one of the most common: TABLE 1 foot is 12 inches, or primes, (') l= I foot 1 inch « 12 seconds, (") 1 second « 12 thirds, rinj TH 1 third * 12 fourths, (c) This of ik of ITT2g&c. forming a decreasing geometrical progression, whose first term is 1, and ratio 12. Hence they are called Duodecimals. 286. How many square feet in a floor, 10ft. 4in. long, and 7ft. Bin, wide ? Here we wish to multiply 10ft. 4. by toft. 44 7ft: ; we therefore write them as at 7 & the left hand, and multiply 4 by 8232; but 4' being of a foot, and 8' 1, the 6 10.8 product is (ms) of a foot, or 72 4 32', which reduced gives 24 81'; put ting down 81', we reserve the 2 to be 79ft gr. Ans. added to the inches. Multiplying foft. by 8=19, the product is (223) #f, to which is being added, we have 3. 10. Next, multiplying 4 by 7==2ft. 4', writing the 4' in the place of inches and reserving the 2ft., we say 7 times 10 are 70, and two added are 72, which we write under the 6ft., and the sum of these partial products is 79ft. 281. Ans. Nutk.-When feet are concerned, the product is of the same denomina siou as the term multiplying the feet; and when feet are uot concerned, the Barns of the product will be denoted by the suur of the indices of the two factory, or stakes over them. Thus, 4 X2= Therefore, DUODECIMALS. 287. To multiply a number consisting of feet, inchus, seconden &c. by another of the same kind. RULE.--Write the several terms of the multiplier under the corresponding terms of the multiplicand; then multiply the whole multiplicand by the several terms of the multiplier saccessively, beginning at the right hand, and placing the first term of each of the partial products under its respective multiplier, remembering to carry one for every 12 from a lower to the next higher denomination, and the sum of these partial products will be the answer, the left hand term being feet, and those towards the right primes, seconds, &c. This is a very useful rule in measuring wood, boards, &c., and for artificers in finding the contents of their work. QUESTIONS FOR PRACTICE. 2. How much wood in a load 8. How many cords in a pile 7ft. 6 long, 4ft. 8' wide, and of 4 foot wood, 24ft. long, and 4ft. high? 6ft. 4' high? Ans. 140ft. or cord 12ft. Ans. 4f cords. Multiply the length by the width, 9. How many square yards and this product by the height. 3. How many square feet in 18ft. long, 16ft. Ở wide, and in the wainscoting of a room a board 16ft. 4in. long, and aft. 9ft. 10 high? Bin. wide ? Ans. 43ft. 6in. 8". -.Ans. 75yd. 3ft. 6. 10. How much wood in a How many feet in a stock cubic pile measuring it on of 12 boards luft. G long, and every side ? Ans. 4 cords. ift. 3 wide.? 11. How many square feet Ans. 217ft. 6'. Nork.-Inches, it will be recol. in a platform, which is 37 feet lected, are so many. 12ths of a foot, 11 inches long, and 23 feet 9 whether the foot is lineal, square, inches broad? or solid. Gin. in the above answer Ans. 900ft. 6 3". is a square foot, or 72 square inches. 12. How much wood in a 5. What is the content of a load 8ft. 4in. long, 3ft. Iin. wide, .ceiling 43ft. 3 long, and 25ft. and 4ft. 5in. high? 6 broad? Ans. 138ft. O 3". Ans. 1102f. 106'. 13. How many feet of floor6. How much wood in a load ing in a room which is 28ft. -6ft. 7 long, 3ft. 5 high, and Gin. long and 23ft. 5ın. broad? 3ft. 8' wide ? Ans. 667ft. 46". Ans. 82ft. 5' 8" 4". 14. How many square feet 7. What is the solid content are there in a board which is of a wall 53ft. 6 long, 12ft. 3 15 feet 10 inches long, and 99 bigh, and 21. thick ? inches wide ? Ans. 13101. I, Alış. 12ft. 10 4" GH!. |