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8. If a sphere of 4 feet in diameter contains 33.5104 cubic
feet, what will be the contents of a sphere 8 feet in diameter ?

4 3 : 8 3 : : 33.5104 : Am.

9. If the contents of a sphere 14 inches in diameter is
1436.7584 cubic inches, what will be the diameter of a sphere
which contains 11494.0672 cubic inches ?

10. If a ball weighing 32 pounds is 6 inches in diameter,
what will be the diameter of a ball weighing 2048 pounds ?

11. If a haystack, 24 feet in height, contains 8 tons of hay,
what will be the height of a similar stack that shall contain
but 1 ton ?

ARITHMETICAL PROGRESSION.

304. An Arithmetical Progression is a series of numbers in
which each is derived from the preceding one by the addition
or subtraction of the same number.

The number added or subtracted is called the common dif-
ference.

305. If the common difference is added, the series is called
an increasing series.

Thus, if we begin with 2, and add the common difference,
3, we have

2, 5, 8, 11, 14, 17, 20, 23, &c.,

which is an increasing series.

If we begin with 23, and subtract the common difference,
3, we hare

23, 20, 17, 14, 11, 8, 5, &c.,
which is a decreasing series.

304. What is an arithmetical progression ? What is the number

305. When the common difference is added, what is the scries called ?
What is it called when the common difference is subtracted ? What
are the several numebrs called ? What arc the first and last called ?
What arc the intermediate ones called ?

ARITHMETICAL PROGRESSION. 291

The several numbers are called the terms of the progres-
sion or series : the first and last are called the extremes, and
the intermediate terms are called means.

306. In every arithmetical progression there are five
parts :

1st, the first term ;

2d, the last term ;

3d, the common difference ;

4th, the number of terms ;

5th, the sum of all the terms.

If any three of these parts are known or given, the remain-
ing ones can be determined.

CASE I.

307. Knowing the first term, the common difference, and
the number of terms, to find the last term.

1. The first term is 3, the common difference 2, and the
number of terms 19 : what is the 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 OPEBATION.
adding the common difference to the 2d ; the 1 8 No. less 1
4th, by adding the common difference to the cj Com dif
3d, and so on ; the number of additions being 1
less than the number of terms found. 35

multiply the common difference by the number ^7: , , ,
of additions, that is, by 1 less than the number m

of terms, and add the first term to the pro-
duct : hence,

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 ivill be the last
term ; if it is decreasing, subtract the product from the
first term and the difference will be the la?t term.

306. How many parts are there in every arithmetical progression ?
What are they ? How many parts must be given before the remaining
ones can be found ?

292 ARITHMETICAL PROGRESSION.

EXAMPLES.

1. A man bought 50 yards of cloth, for which he was tQ
pay 6 cents for the 1st yard, 9 cents for the 2d, 12 cents for
the 3d, and so on increasing by the common difference 3 :
how much did he pay for the last yard ?

2. A man puts out \$100 at simple interest, at 1 per cent :
at the end of the 1st year it will have increased to \$107, at
the end of the 2d year to \$114, and so on, increasing \$t
each year : what will be the amount at the end of 1 6 years ?

3. What is the 40th term of an arithmetical progression of
which the first term is 1, and the common difference 1 ?

4. What is the 30th term of a descending progression of
which the first term is 60, and the common difference 2 ?

5. A person had 35 children and grandchildren, and it so
happened that the difference of their ages was 18 months,
and the age of the eldest was 60 years : how old was the
youngest ?

CASE II.

308. Knowing the two extremes and the number of terms,
to find the common difference.

1. The extremes of an arithmetical progression are 8 and
104, and the number of terms 25 : what is the common dif-
ference ?

ANALYSIS. Since the common difference
multiplied by 1 less than the number of OPERATION.

terms gives a product equal to the differ 104

erence of the extremes, if we divide the dif g

ference of the extremes by 1 less than the

number of terms, the quotient will be the 25 1 24)96(4.
common difference : hence,

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.

307. "When you know the first term, the common difference, and the
number of terms, how do you find the last term ?

308. When you know the extremes and the number of terms, how do
you find the common difference ?

ARITHMETICAL PROGRESSION. 293

EXAMPLES.

1. A man has 8 sons, the youngest is 4 years old and the
eldest 32 : their ages increase in arithmetical progression :
what is the common difference of their ages ?

2. A man is to travel from New York to a certain place in
12 days ; to go 3 miles the first day, increasing every day by
the same number of miles,; the last day's journey is 58 miles :
required the daily increase.

3. A man hired a workman for a month of 26 working
days, and agreed to pay him 50 cents for the first day, with
a uniform daily increase ; on the last day he paid \$1.50 :
what was the daily increase ?

CASE III.

309. To find the sum of the terms of an arithmetical
progression.

1. What is the sum of the series whose first term is 3,
common difference 2, and last term 19 ?
Given scries - 3+ 5 + 1 + 9 + 11 + 13 + 15 + 17 + 19= 99

ofTcnnshv-l 19 + 17 + 15 + 13 + 11+ 9+ t+ 5+ 8= 99

verted. J

Sura of both. 2'2 iJ 22 22 22 22 22 22 22 198

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 3 and 19 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 the series.

EXAMPLES.

1. The extremes are 2 and 100, and the number of terms
22 : what is the sum of the series?

OPERATION.

ANALYSIS. We first add 2 1st term,

together the two extremes inn lost tpvm
and then multiply by half la
the number of terms. 1 02 sum of extremes.

11 half the number of terms

1122 sum of series.
309. How do you find the sum of the terms?

294 GEOMETRICAL PKOGEESSION.

2. How many strokes does the hammer of a clock strike iu
12 hours?

3. The first term of a series is 2, the common difference 4,
end the number of terms 9 : what is the last term and sum of
the series ?

4. James, a smart chap, having learned arithmetical pro-
gression, told his father that he would chop a load of wood of
15 logs, at 2 cents for the first log, with a regular increase of
1 cent for each additional log : how much did James receive
for chopping the wood ?

5. An invalid wishes to gain strength by regular and in-
creasing exercise ; his physician assures him that he can
walk 1 mile the first day, and increase the distance half a
mile for each of the 24 following days : how far will he
walk ?

C. If 100 eggs are placed in a right line, exactly one yard
from each other, and the first one yard from a basket : what
distance will a man travel who gathers them up singlv and
places them in the basket ?

GEOMETRICAL PROGRESSION.

310. A GEOMETRICAL PROGRESSION is a series of terms,
each of which is derived from the preceding one, by multi-
plying it by a constant number. The constant multiplier is
called the ratio of the progression.

311. If the ratio is greater than 1, each term is greater
than the preceding one, and the series is said to be in-
creasing.

31.0. What is a geometrical progression? What is the constant
multiplier called ?

311. If the ratio is greater than 1, how do the terms compare with
each other? What is the series then called? If the ratio is less
than 1, how do they compare ? What is the series then called ? What
arc the several numbers called? What are the first and last called?
What are the intermediate ones called ?

312. How many parts are there in every geometrical progression ?
What are they? How manv must be known before the others can be
found ?

GEOMETRICAL PROGRESSION. 295

If the ratio is less than 1, each term is less than the
preceding one, and the series is said to be decreasing;
thus,

1, 2, 4, 8, 16 ; 32, &c. ratio 2 increasing series :
32, 16, 8, 4, 2, 1, &c. ratio 1 decreasing series.

The several numbers are .called terms of the progression.
The first and last are called the extremes, and the intermedi-
ate terms are called means.

312. In every Geometrical, as well as in every Arithmeti-
cal Progression, there are five parts :

1st, the first term ;
2d, the last term ;
3d, the common ratio ?
4th, the number of terms ;
5th, the sum of all the terms.

If any three of these parts are known, or given, the re-
maining ones can be determined.

CASE I.

313. Having given the first term, the ratio, and the
number of terms, to find the last term.

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

ANALYSIS. The se- OPERATION.

cond term is formed by 2x2x2x2x 2=:2 5 = 32
multiplying the first 3 j t t

term by the ratio ; tho _____

third term by multiply- Ans. 96

ing the second term by

the ratio, and so on ; the number of multiplications being 1 less
iJian the number of terms : thus,

3 3 1st term,

3x2 = 6 2d term, *

3x2x2=3x2-=12 3d term,

3 x 2 x 2 x 2^3 x 2 3 24 4th term, <fcc.

296 GEOMETRICAL PROGRESSION.

Therefore, the last term is equal to the first term multi-
plied by the ratio raised to a power 1 less than the number
of terms.

RULE. Eaise the ratio to a power whose exponent is 1
less than the number of terms, and then multiply this power
by the first term.

EXAMPLES.

1. The first term of a decreasing progression is 192 ; the
ratio i, and the number of terms 7 : what is the last term ?

NOTE. The 6th power of the ratio, (-), is OPERATION.

^4, and this multiplied by the first term 192, (l) 6 = Jk-

gives the last term 3. 1 92 X -^-=3

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

3. The first term of a decreasing progression is 1024, the
ratio i : what is the 9th term ?

4. The first term of an increasing progression is 4, and the
common ratio 3 : what is the 10th term ?

5. 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 ?

6. A man bought 12 yards of cloth, giving 3 cents for the
1st yard, 6 for the 2d, 12 for the 3d, &c. : what did he pay
for the last yard ?

CASE II.

314. Knowing the two extremes and the ratio, to find
the sum of the terms.

1. What is the sum of the terms in the progression, 1, 4,
16, 64 ?

313. Knowing the first term, the ratio, :ind the number- of terms, 1 row-
do you find the Itust term ?

314. Knowing the two extremes and the ratio, how do you find the
sum of the terms V

GEOMETRICAL PROGRESSION. 297

ANALYSIS. If we multiply the terms of the progression by the
Tatio 4, we have a second pro-
gression, 4, 16, 64, 256, which OPERATION.
is 4 times as great as the first. 4+16+64+256= 4 times.

If from this we subtract the 1+4+16+64 =_ once.

first, the remainder, 2561, 256 1=3 times.

will be 3 times as great as 9 ~/ 1 9 ~,-

the first; and it the remain- !== - = 85 sum.

der be divided by 3, the quo- '

tient will be the sum of the

terms of the first progression. But 256 is the product of the last

term of the given progression multiplied by the ratio, 1 is the first

term, and the divisor 3 is 1 less than the ratio ; hence,

RULE. Multiply the last term by the ratio ; take the dif-
ference between the product and the first term and divide
the remainder by the difference between 1 and the ratio.

NOTE. When the progression is increasing, the first term is
subtracted from the product of the last term by the ratio, and the
divisor is found by subtracting 1 from the ratio. When the pro-
gression is decreasing, the product of the last term by the ratio is
subtracted from the 'first term, and the ratio is subtracted from 1.

EXAMPLES.

1. The first term of a progression is 2, the ratio 3, ami the
last term 4374 : what is the sum of the terms ?

2. The first term of a progression is 128, the ratio J, and
the last term 2 : what is the sum of the terms ?

3. The first term is 3, the ratio 2, and the last term 192 :
what is the sum of the series ?

4. A gentleman gave his daughter in marriage 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 ?

5. A man bought 10 bushels of wheat*on the condition that
he should pay 1 cent for the 1st bushel, 3 for the 2d, 9 for
the 3d, and so on to the last : what did he pay for the last
bushel, and for the 10 bushels?

6. A man has 6 children : to the 1st he gives \$150, to the
2d \$300, to the 3d \$600, and so on, to each twice as much
as the last : how much did the ehKst, Teceive, and what was
the amount received by them all ?

298 PROMISCUOUS QUESTIONS.

PROMISCUOUS EXAMPLES.

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

2. How many bushels of oats at 62^ cents a bushel will pay
for 4250 feet of lumber at \$7.50 per thousand ?

3. Bougkt 27ihd. of sugar which weighed as follows : the 1st
5cwt. Iqr. ISlb., the 2d Gcwt. IQlb. : what did it cost at 7 cents per
pound?

4. How many hours between the 4th of Sept., 1854, at 3 P.M.,
and the 20th day of ApriJ, 1855, at 10 A.M. ?

5. If | of a gallon of wine cost of a dollar, what will - of a

6. What number is that which being multiplied by \ will pro-
duce i?

7. A tailor had a piece of cloth containing 24 yards, from which
he cut 6 1 yards : how much was there left ?

8. From | offtake lof^'

9. What is the difference between 3| + 7| and 4 + 2-H ?

10. There was a company of soldiers, of whom \ were on guard,
preparing dinner, and the remainder, 85 men, were drilling :

ow many were there in the company ?

11. The sum of two numbers is 425, and their difference 1.625:
what are the numbers ?

12. The sum of two numbers is f, and their difference ^ : what
are the numbers ?

13. The product of two numbers is 2.26, and one of the numbers
is .25 : what is the other ?

14. If the divisor of a certain number be 6.66, and the quo-
tient \ , what will be the dividend ?

15. A person dying, divided his property between his widow and
his four sons ; to his widow he gave \$1780, and to each of his
on an average 126 dollars a year : how much had he when he

16. A besieged garrison consisting of 360 men was provisioned
for 6 months, but hearing of no relief at the end of five months,
dismissed so many of the garrison, that the remaining provision
lasted 5 months : how many men were sent away ?

17. 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 the amount of their indebtedness ?

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

PROMISCUOUS QUESTIONS. 299

19. How many planks 15 feet long, and 15 inches wide, will
floor a barn 60^ feet long, and 33i feet wide?

20. A person owned f of a mine, and sold f of his interest for
\$ 1710 : what was the value of the entire mine ?

21. A room 30 feet long, and 18 feet wide, is to be covered with
painted cloth f of a yard wide : how many yards will cover it ?

22. 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.

23. A can do a piece of work in 12 days, and B can do the same
work in 18 days : how long will it take both, if they work together?

24. If a barrel of flour will last one family 7 months, a second
family 9 months, and a third ll months, how long will it last the
"three families together ?

25. Suppose I have -,% of a ship worth \$1200 ; what part have
I left after selling | of \$ of my share, and what is it worth?

26. What number is that which being multiplied by of f of
1 , the product will be 1 ?

27. Divide \$420 between three persons, so that the second shall
have f as much as the first, and the third ^ as much as the other two ?

28. What is the difference between twice five and fifty, and
twice fifty five ?

29. What number is that which being multiplied by three-
thousandths, the product will be 2637 ?

30. What is the difference between half a dozen dozens and six
dozen dozens?

31. The slow or parade step is 70 paces per minute, at 28 inches
each pace : how fast is that per hour ?

32. 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 ?

33. A wall of 700 yards in length was to be built in 29 days :
12 men were employed on it for 11 days, and only completed 220
yards : how many men must be added to complete the wall in the
required time ?

34. Divide \$10429.50 between three persons, so that as often
as one gets \$4, the second will get \$6 and the third \$7.

35. A gentleman whose annual income is 1500, spends 20
guineas a week ; does he save, or run in debt, and how much ?

36. A farmer exchanged 70 bushels of rye, at \$0.92 per bushel,
for 40 bushels of wheat, at \$1.874/ a bushel, and received the
balance in oats, at \$0.40 per bushel : how many bushels of oats

37. In a certain orchard of the trees bear apples, i of them
bear peaches, of them plums, 120 of them cherries, and 80 of
them pears: how many trees are there in the orchard ?

300 PKOMISCUOUS QUESTIONS.

38. A person being asked the time, said, the time past noon
is equal to of the time past midnight : what was the hour ?

89. If 20 men can perform a piece of work in 12 days, how
many men will accomplish thrice as much in one-fifth of the time?

40. How many stones 2 feet long, 1 foot wide, and 6 inches
thick, will build a wall 12 yards long, 2 yards high, and 4 feet
thick ?

41. Four persons traded together on a capital of \$6000, of
which A put in , B put in ^, C put in %, and D the rest ; at the
end of 4 years they had gained \$4728 : what was each one's share of
the gain ?

42. A cistern containing 60 gallons of water has three unequal
pipes for discharging it ; the largest will empty it in one hour, the
second in two hours, and the third in three hours : in what time
will the cistern be emptied if they run together ?

43. A man bought f of the capital of a cotton factory at par ;
he retained of his purchase, and sold the balance for \$5000
which was 15 per cent advance on the cost ; what was the whole
capital of the factory ?

44. Bought a cow for \$30 cash, and sold her for \$35 at a credit
of 8 months : reckoning the interest at 6 per cent, how much did
I gain ?

45. If, when I sell cloth for 8-?. Qd. per yard, I gain 12 per cent,
what per cent will be gained when it is sold for 10s. Qd per yard ?

46. How much stock at par value can be purchased for \$8500,
at 8^ per cent premium, per cent being paid to the broker?

47. Twelve workmen, working 12 hours a day, have made in
12 days, 12 pieces of cloth, each piece 75 yards long ; how many
pieces of the same stuff would have been made, each piece 25
yards long, if there had been 7 more workmen ?

48. A person was born on the 1st day of Oct., 1801, at 6 o'clock
in the morning, what was his age on the 21st of Sept., 1854, at
half-past 4 in the afternoon?

49. A, can do a piece of work alone in 10 days, and B in 13
days : in what time can they do it if they work together?

50. A man went to sea at 17 years of age; 8 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 ?

51. How many bricks, 8 inches long and 4 inches wide, will
pave a yard that is 100 feet by 50 feet ?

52. If a house is 50 feet wide, and the post which supports the
ridge pole is 12 feet high, what will be the length of the rafters?

53. 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 ?

PROMISCUOUS QUESTIONS. 301

54. If a quantity of provisions serves 1500 men 12 weeks, at
the rate of 20 ounces a day for each man, how many men will the
same provisions maintain for 20 weeks, at the rate of 8 ounces a
day for each man ?

55. A man bought 10 bushels of wheat, on the condition that
he should pay 1 cent for the 1st bushel, 3 for the 3d, 9 for the 3d,
and so on to the last : what did he pay for the last bushel, and for
the 10 bushels ?

56. There is a mixture made of wheat at 4s. per bushel, rye at
3s., barley at 2s., with 12 bushels of oats at 18d. per bushel : how
much must be taken of each sort to make the mixture worth 2s.,
tid. per bushel ?

57. What length must be cut off a board 8^ inches broad to
contain a square foot ?

58. What is the difference between the interest of \$2500 for 4
years 9 mo. at 6 per cent., and half that sum for twice the time,
at half the same rate per cent ?

59. A person lent a certain sum at 4 per cent, per annum ; had
this remained at intera Bt 3 years, he would have received for prin-
cipal and interest \$9676.80 : what was the principal?

60. If: 1 pound of tea be equal in value to 50 oranges, and 70
oranges be worth 84 lemons, what is the value of a pound of tea,
when a lemon is worth 2 cents ?

61. A person bought 160 oranges at 2 for a penny, and 180
more at 3 for a penny ; after which he sold them out at the rate
of 5 for 2 pence .did he make or lose, and how much ?

62. 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 ?

63. A ship has a leak by which it would fill and sink in 15
hours, but by means of a pump it could be emptied, if full, in
16 hours. Now, if. the pump is worked from the time the leak
begins, how long before the ship will sink ?

64. A and B can perform a certain piece of work in 6 days, B
and C in 7 days, and A and C in 14 days : in what time would
each do it alone ?

65. Divide \$500 among 4 persons, so that when A has i dollar
B shall have , C, |, and D .

66. A man purchased a building lot containing 3600 square
feet, at the cost of \$1.50 per foot, on which he built a store at an
expense of \$3000. He paid yearly \$180.66 for repairs and taxes :
what annual rent must he receive to obtain 10 per cent on the
cost?

67. A's note of \$7851.04 was dated Sept. 5th, 1837, on which
were endorsed the following payments, viz. : Nov. 13th, 1839,
\$416.98; May 10th, 1840, \$152- what was due March 1st, 1841,
the interest being 6 per cent ?

302 PROMISCUOUS QUESTIONS.

68. A Louse is 40 feet from the ground to the caves, and it is
required to find the length of a ladder which will reach the eaves,
supposing the foot of the ladder cannot be placed nearer to the
house than 30 feet ?

G9. Sound travels about 1142 feet in a second ; now, if the
flash of a cannon be seen at the moment it is fired, and the report
heard 45 seconds after, what distance would the observer be from
the gun ?

70. A person dying, worth \$5460, left a wife and 2 children, a
son and daughter, absent in a foreign country. He directed that

" if his son returned, the mother should have one third of the estate
and the son the remainder ; but if the daughter returned, she

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