RULE.

Raise the ratio to a power whose exponent is 1 less than the number of terms, and then multiply the power by the first term, the product will be the last

term.

Ex. 1. The first term is 3 and the ratio 2; what is the 6th term?

2×2×2×2×2-25=64

3 1st term

Ans. 192

2. A man purchased 12 pears: he was to pay 1 farthing for the first, 2 farthings for the 2d, 4 for the 3d, and so on, doubling each time: what did he pay for the last?

Ans. £2 2s 8d.

3. A gentleman dying left nine sons, and bequeathed his estate in the following manner to his executors £50; his youngest son to have twice as much as the executors, and each son to have double the amount of the son next younger : what was the eldest son's portion?

CASE II.

Ans. £25600.

§ 231. Having given the two extremes to find the sum of the series.

RULE.

Subtract the less extreme from the greater, divide the remainder by 1 less than the ratio, and to the quotient add the greater extreme: the sum will be the sum of the series.

r. 1. The first term is 3, the ratio 2, and the last 22: what is the sum of the series?

192-3-189 difference of the extremes, =1)189(189; then 189+ 192-381 Ans.

2. A gentleman married his daughter on New Year's day, and gave her husband Is towards her portion, and was to double it on the first day of every month during the year: what was her portion? Ans. £204 15s.

QUESTIONS.

223. How do you form an Arithmetical Series? What is the common difference? What is an ascending series? What a descending series? What are the several numbers called? What are the first and last terms called? What are the intermediate terms called?

§ 224. In every Arithmetical Progression how many things are considered? What are they?

§ 225. How do you find the last term when the first term and common difference are known?

§ 226. How do you find the common difference, when you know the two extremes and number of terms?

227. How do you find the sum of the series?

228. How do you form a Geometrical Progression? What is the common ratio? What is an ascending series ?-What a descending series? What are the several numbers called? What are the first and last terms called? What are the intertermediate terms called?

§ 229. How many things are considered in a Geometrical Progression? What are they?

230. How do you find the last term when the first term and common difference are known?

231. How do you find the sum of the series?

PROMISCUOUS QUESTIONS.

1. A merchant bought 13 packages of goods, for which he paid $326: what will 39 packages cost at the same rate?

Ans. $978.

2. Two merchants, A and B traded together; A put in £320 for 5 months, and B £460 for 3 months;

they gained £100: how much should each one receive?

Ans. A £53 13s 947fd, B £46 6s 24 d. 3. If I buy 1000 Ells Flemish of linen, for £90 what must it be sold for per Ell English, to make £10 by the purchase? Ans. 3s 4d.

4. What number taken from the square of 54 will leave 19 times 46? Ans. 2042. 5. If of a gallon of wine cost of a £, what wil of a tun cost?

Ans. £105. 6. If an officer's salary be £48 per annum, how much will he receive in 232 days?

Ans. £30 10s 2‡d+. 7. If a gentleman spends, one day with another, £1 7s 101d, and at the end of the year has saved £340, what is his yearly income?

Ans. £848 14s 44d.

8. If 8 cannons expend, in one day, 48 barrels of powder, how much will 24 cannons expend in 22 days?

Ans. 3168bar.

9. What number is that which being multiplied by will produce?

Ans. 3.

10. A person dying divided his property between his widow and his four sons: to his widow he gave $1780, and to each of his sons $1250: he had been 25 years in business, and had cleared on an average $126 a year: how much had he when he began business? Ans. $3567. 11. A besieged garrison consisting of 380 men isioned for 6 months, but hearing of no red of 5 months, dismissed so many of the

garrison that the remaining provision lasted 5 months: how many men were sent away?

Ans. 288.

12. What number added to 114, will produce 36311?

Ans. 24813.

13. What number multiplied by, will produce 11?

Ans. 261f.

14. A man had 12 sons, the youngest was 3 years old and the eldest 58, and their ages increased in arithmetical progression: what was the common difference of their ages?

Ans. 5 years. 7

15. A snail in getting up a pole, 20 feet high, was observed to climb up 8 feet every day, but to descend 4 feet every night: in what time did he reach the top of the pole?

Ans. 4 days.

16. Two persons, A and B are indebted to C; A owes $2173, which is the least debt, and the difference of the debts is $371: what is B's debt?

Ans. $2544,

17. What is the difference between twice four and forty and twice forty-four: also between twice five and fifty and twice fifty-five?

Ans. 40 and 50.

18. A lady being asked her age, and not wishing to give a direct answer, said, I have 9 children, and three years elapsed between the birth of each of them; the eldest was born when I was 19 years old, and the youngest is now exactly 19: what was her age? Ans. 62 years.

19. What number added to the 43d part of 4429, will make the sum 240?

Ans. 137.

8

20. A man went to sea at 17 years of age; years after he had a son born, who lived 46 years, and died before his father: after which the father lived twice twenty years and died: what was the age of the father? Ans. 111 years.

21. A brigade of horse consisting of 384 men, is to be formed into a solid body consisting of 32 men in front; how many ranks will there be ?

Ans. 12. 22. A room 30 feet long and 18 feet wide is to be covered with painted cloth of a yard in width: how many yards will cover it?

Ans. 80.

23. A, B and C trade together and gain $120, which is to be shared according to each one's stock; A put in $140, B $300, and C $160: what is each man's share?

Ans. A's $28, B's $60, and C's $32. 24. There is a stone which measures 4 feet 6 inches long, 2 feet 9 inches broad, and 5 feet 4 inches deep: how many solid feet does it contain ?

Ans. 66 feet.

25. Two men depart from the same place and travel in different directions; one goes 7 miles and the other 11 miles per day: how far will they be apart at the end of the 12th day?

Ans. 216 miles. 26. How many planks 15 feet long and 15 inches wide, will floor a barn 60 feet long and 334 feet wide? Ans. 1087.

97. A person owned of a mine, and sold † of his st for $1710: what was the value of the enine? Ans. $3800.