EXAMPLE S. 1. A person takes out of his pocket, at 10 different. times, fo many different numbers of crowns, every one exceeding the former by 2; the laft was 23; what was the firft? 10-1=9x2=18. Then 23-18-5 the Ani. 2. A traveller performs a journey in 20 days, and makes every day's journey 4 miles greater than that preceding. His laft day's journey was 59 miles; how far did he travel the first day? Anf. 17 miles. 3. The leaft term is 3, the common difference 2, and the number of terms 9; what is the greatest term? Anf. 19. 4. A man fold his coat after the rate of is. for the firft button, 28. for the fecond, 3s. for the third, and fo on to the whole number of buttons upon it, which was 12; what was the coat fold for?-Anf. 31. 18s. One extreme, the number of terms, and the sum of the feries being given, to find the other extreme. R U L E. Multiply the fum of the feries by 2, and divide the product by the number of terms; from this quotient fubtract the given extreme, and the remainder will be the extreme fought. EXAMPLE S. 1. A gentleman received 420l. at 12 feveral payments in Arithmetical progreffion; the firft payment was 31. 8s. required the laft payment?. £. ko. £. s. £. s. 420X2=840÷12=70—3 8=66 12 Anf. 2. The greatest term is 45, the number of terms 13, and the sum of the feries 325; what is the leaft term? Anf. 5. GEO. GEOMETRICAL PROGRESSION. GEometrical progreffion is when any series or rank of numbers increase or decrease by a common multiplier or divifor, called the ratio; as, 2, 4, 8, 16, 32 increase by the common ratio 2; and 81, 27, 9, 3, 1 decrease by the common ratio 3. N OT E. ift. In Geometrical progreffion, the fame 5 things are to be observed as in Arithmetical progreffion, viz. ft. The firft term. 2d. the last term. 3d. the number of terms. 4th. the ratio. 5th. the fum of all the terms. Any three of which being known, the rest may be found. zd. The principal difference between Arithmetical progreffion, and Geometrical progreffion confifts in ufing addition, fubtraction, multiplication, and divifion in the former, and multiplication, divifion, involution and evolution in the latter. The rules for both are nearly the fame. 3d. The remark in Arithmetical progreffion is equally applicable to this rule. The firit term, the last term, and the ratio being given, to find the sum of the feries. R U L E. Multiply the laft term by the ratio, and from the product fubtract the firft term, and the remainder divide by the ratio lefs 1 for the fum of the series. EXAMPLE S. 1. If the extremes be 5 and 885735, and the ratio 3; what is the fum of the feries? 885735×3=2657205-5=2657200÷2(—3—1)= 1328600 Anf. 2. The extremes of a Geometrical feries are 1024 and 59949, and the ratio is 1; what is the fum of the feries? Anf. 175099. 3. On new year's day a gentleman married, and received of his father-in-law a guinea, with a promise that he was to have a prefent, on the first day of every month, double the preceding one, for the firft year ; what would the lady's portion be? Seffa Anf. 42991. 158. 4. It is faid that Seffa, who invented the game of chefs, was bid by his prince to name his own reward. afked 1 grain of wheat for the firft fquare on the chefsboard, 2 grains for the fecond, 4 for the third, and fo on, doubling the grains for each following fquare, to the whole number on the board, which is 64. Now, allowing 7680 grains to a pint, how many bushels of wheat will his reward amount to? Anf. 37529996894754 bls. 1 gn. Opts. 4095 qrs.. 5. A laceman well verfed in numbers, agreed with a gentleman to fell him 20 yards of rich gold brocaded lace, for 2 pins the first yard, 6 for the fecond, 18 for the third, and fo on in triple proportion. I demand how much the lace produced, fuppofing the pins worth a farthing per 100; alfo whether the laceman gained or loft by the fale thereof, the lace having coft him 81. Is. 8d. per yard? £. S. d. 290565 7 4 290403 14 O II. PROBLEM One of the extremes, the ratio, and the number of terms being given, to find the other extreme. RULE. From the number of terms subtract 1, and involve the ratio to the power expreffed by the remainder. Then, multiply or divide the given extreme by the said power, according as it is the greateft or leaft, and the product or quotient will be the extreme fought. E X AM P L E S. 1. The greatest term is 512, the ratio 2, and the number of terms 10; what is the least term? 10-19, and 2×2X2X2×2×2X2X2X2=512÷512 = the Anf. 2 A boy agrees for 16 oranges, to pay only the price of the laft, reckoning a farthing for the firft, a half penny for the fecond, &c. doubling the price to the laft. How much did he pay for the whole bargain? Anf. 341. 2s. 8d. 3. What debt will be difcharged in a year, or 12 months, by paying 11. the firft month, 21. the fecond,. 41. the third, and fo on; and what will the last ment be.?. pay £. Anf. 4095 the debt. III. 1 PROBLEM The firft term, the laft term, and the fum of the feries being given, to find the ratio. R U Ľ E. From the fum of the feries fubtract the first term, and divide the remainder by the difference between the faid fum and the last term, and the quotient will be the ratio required. Z 3 EX EXAMPLE S. 1. The extremes are 5 and 885735, and the fum of the feries is 1328600; what is the ratio? 1328600-5=1328595, and 1328600-885735442865. Then 1328595÷442865=3 the Anf. I 2. If the extremes of a series be 1 and 729, and the fum of the series 1093; what is the ratio? Anf. 3. 3. A man received 4095 guineas in one year, by monthly payments in Geometrical progreffion, and his first receipt was I guinea, and his laft 2040 guineas; in what ratio was the money received? Anf. 2. The firft term and ratio of a decreafing feries being given, to find the fum of the feries. R U L E.. Divide the fquare of the first term by the difference between the first and fecond term of the feries, and the quotient will be the answer. EXAMPLE S. 1. Required the fum of the infinite Geometrical series 8, 4, 2, 1, 1, &c.? 8X8=64÷4 (8-4) 16 the Anf. 2. Suppose a ball flies 12 miles the firft hour, 10 the fecond, and fo on in the proportion of 12 to 10 to infinity; what space would it move through? Anf. 72 miles. 3. What is the fum of the infinite feries, T r, &c.? Anf.. POSITION |