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One pound of prepared white lead paint will cover of first coat about 4 square yards, and of subsequent coats from 41 to 54 square yards.

A day's work for a painter is, of plain outside work, from 80 to 100 square yards, and of inside work, from 40 to 65 square yards.

87. Each of the two sides of the roof of a certain building is 34 feet long and 25 feet wide. How many shingles will be required for it if laid 4 inches to weather, and how many 4-penny nails must be allowed ? What will be the cost of laying the shingles, at $ 2.50 per day for labor ?

88. How many clapboards, laid 4 inches to the weather, will be required to cover the side of a building 30 feet high and 63 feet long, no allowance being made for openings ?

89. A close board fence, 4 feet high and 64 feet long, is to be painted both sides with two coats of white lead paint. When the lead, ground in oil, is 9 cents a pound, and linseed oil 72 cents a gallon, what will the paint required cost? How much must be paid a painter for putting it on, if he covers 80 square yards a day at $ 2.50 ?

90. A room is 20 feet long, 18 feet wide, and 10 feet high. Allowing 108 square feet for openings and mopboards, how many laths will be required for its ceiling and walls ? What will be the cost of nails for setting them at 44 cents a pound ?

91. A hipped-roof barn is 64 feet long, 40 feet wide, and 20 feet high to the roof. Allowing 360 square feet for openings, how much will rough boards for the sides cost at $ 20 a thousand feet, and how much should each of two men, at $ 2.25 a day, be paid for putting on the boards ?

PROGRESSIONS.

539. A Series of numbers is a succession of numbers, increasing or decreasing according to some fixed law.

540. The Terms of a series are the numbers forming the series.

The first and last terms are called the extremes, and the intervening terms the means. Thus,

3, 6, 9, 12, is a series in which 3 and 12 are the extremes, and 6 and 9 the means.

541. A series is ascending or descending, according as the séries increases or decreases from the first term.

ARITHMETICAL PROGRESSION.

542. An Arithmetical Progression is a series of numbers which increase or decrease by a common difference. Thus,

2, 4, 6, 8, 10, 12, is an ascending series,
12, 10, 8, 6, 4, 2, is a descending series,

in each of which 2 is the common difference.

To find either Extreme.

92. The first term of an arithmetical progression is 3, and the common difference 2. What is the fifth term ?

Solution.

1st term : 3.

3d term = 3+ (2 X 2).
2d term =
3+ 2.

4th term=3+ (2 X 3).
5th term = 3+ (2 X 4) = 11.

93. The fifth term of an arithmetical progression is 11, and the common difference 2. What is the first term ?

Solution.

(2 X 2).
(2 X 3).

5th term = ll.

3d term = 11 – 4th term = 11 - 2.

2d term = 11

1st term =ll — (2 X 4) = 3. Hence,

543. To find an extreme,

Multiply the common difference by the number of terms less one, and the product plus the smaller extreme will be the larger ; or, the larger extreme minus the product will be the smaller

94. The number of terms of an arithmetical progression is 100, the common difference 3, and the first term 5. What is the last term ?

95. A man bought 34 yards of cloth, and agreed to give 12 cents for the first yard, 12} cents for the second yard, 123 cents for the third yard, and so on. What did the last yard cost him ?

96. A man travels 10 days, increasing each day's travel by

of a mile. If he goes the last day 17 miles, how many miles did he start with ?

97. If 16 persons give in charity, and the first gives 5 cents, the second 9 cents, and so on in arithmetical progression, how much does the last person give ?

To find the Sum of the Terms.

98. The first term of an arithmetical progression is 2, the last term 12, and the number of terms 6. What is the sum of all the terms ?

Solution. Let 2,

4, 6, 8, 10, 12, be an arithmetical series, and 12, 10, 8, 6, 4, 2, be the series reversed.

14 + 14 + 14+ 14 + 14 +1+= 84, twice the sum of the series,

But 84 = (2 + 12) X 6, or the sum of the extremes multiplied by 6 ; and half of 84, or

(2 + 12) X 6

= the sum of the series.

2 Hence,

544. To find the sum of the terms,

Multiply the sum of the extremes by the number of terms, and take half the product.

99. The first term of a series is 2, the last term 478, and the uumber of terms 86. What is the sum of the series ?

100. A man agreed to labor 12 months. For the first month he was to be paid $ 7, and for the last $51. If he was to receive the same addition to his wages each successive month, what sum would he receive for his year's labor ?

GEOMETRICAL PROGRESSION.

545. A Geometrical Progression is a series of numbers which increase, or decrease, by a common rate or ratio. Thus,

3,9, 27, 81, 243, is an ascending series ;

243, 81, 27, 9, 3, is a descending series. In the first series the rate, or ratio, is 3, and in the last }.

To find any Term. 101. The first term of a geometrical progression is 4, the rate 2, and the number of terms 5. What is the last term ?

Solution. 1st term = 4.

3d term = 4 x 22. 2d term = 4 X 2.

4th term 5th term = 4 X 2 = 64.

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102. The first term of a geometrical progression is 1458, and the rate }. What is the seventh term ?

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546. To find any term,

Multiply the first term by that power of the ratio whose exponent is equal to the number of terms less one.

103. The first term of a series is 10, the rate 20, and the number of terms 5. What is the last term ?

104. When the first term is $ 120, the ratio 1.06, and the number of terms 4, what is the last term ?

105. What will $ 50 amount to in 4 years at 6 % compound interest ?

To find the Sum of the Series. 106. A geometrical progression consists of 2, 6, 18, 54, the ratio being 3. What is the sum of all the terms ?

Solution.
6 + 18 + 54 + 162 = 3 times the series.
2 + 6 + 18 + 54 = once the series.
162 2

= 2 times the series.
162 2

= 80 = the series.

2 Subtracting like terms of once the series from three times the series, there remains 162 · 2, as two times the series, or 80 as the sum of the series. Hence,

547. To find the sum of the series,

Multiply the last term by the ratio, subtract the first term, and the remainder divided by the ratio less one will give the

sum.

107. What is the sum of a geometrical progression whose extremes are 6 and 768, and ratio 2 ?

108. The first term of a geometrical progression is 10, the ratio 4, and the number of terms 5. What is the sum of the series to the nearest hundredth ?

109. The first term of a geometrical series is $ 100, the rate 1.06, and the number of terms 4. What is the sum of the series ?

110. A lady, wishing to purchase 10 yards of velvet, thought $4 a yard too high a price. She, however, agreed to give 1 cent for the first yard, 4 cents for the second, 16 cents for the third, and so on. What was the cost of the velvet ?

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