13. A composition was made of 5 lbs. of tea at D11 per lb., 9 lbs. at D1.80 per lb., and 17 lbs. at D1 per lb.; what is a pound of it worth? Ans. D1.54.6.

14. A grocer would mix different quantities of sugar, namely one at 20c., one at 23c., and one at 26c.; what quantity of each sort must be taken to make a mixture worth 22c. per lb. ? Ans. 5 lbs. at 20c., 2 at 23c., and 2 at 26c.

Demonstration.-By connecting the less rate with the greater, and placing the difference between them and the mean rate alternately, or one after the other, in turn; the quantities are such, that there is precisely as much gained by one quantity as is lost by the other, and therefore the gain and loss, upon the whole, are equal, and exactly the proposed rate. In like manner, let the number of simples be what it may, and with how many soever each one is linked, since it is always a less with a greater than the mean price, there will be an equal balance of loss or gain between every two, and consequently an equal balance on the whole. The rule is founded on the principles of proportion.

REVIEW.

What is Alligation? What is Alligation Medial? How do you find the price of the mixture? What is the rule? What is Alligation Alternate? How can you prove it? How do you find the proportional parts when the price only is given? Repeat rule 1st. What is the rule when a given quantity of one of the simples is to be taken?

What is the rule when the quantity of the compound as well as the price is given? What more can you say of Alligation? 15. Bought a pipe of wine, containing 120 gallons, at D1.30 a gallon; how much water must be mixed with it to reduce the first price to D1.00 a gallon? Ans. 21 gallons.

ARITHMETICAL PROGRESSION.

ANY series of numbers more than two, increasing or decreasing by an even ratio or common difference, is in Arithmetical Progression. When the numbers are formed by continual addition of the ratio or common difference, they form an ascending series; but when formed by continual subtraction of the common difference, it is a descending series.

Thus { 0, 2, 4, 6, 8, 10, &c., is an ascending arithmetical series.

And

1, 2, 4, 8, 16, 32,
10, 8, 6, 4, 2, 0,
32, 16, 8, 4, 2, 1,

is an ascending geometrical series. is a descending arithmetical series. is a descending geometrical series.

The numbers which form the series are called the terms of

the progression.

*

The first and last terms of a progression are called the extremes, and the other terms the means. Any three of the fol lowing things being given, the other two may be easily found 1st, the first term; 2d, the last term; 3d, the number of terms; 4th, the common difference; 5th, the sum of all the terms.

The first term, the last term, and the number of terms being given, to find the common difference.

RULE 1.

Divide the difference of the extremes by the number of terms, less 1, and the quotient will be the common difference sought. 1. The extremes are 3 and 39, and the number of terms is 19; what is the common dfference?

Extremes.

39-3=36÷19-1-18, or 18)36(2 Ans.

2. A man had 10 sons whose several ages differed alike; the youngest was 3 years old, and the eldest 48; what was the common difference of their ages? Ans. 5 years.

The first term, the last term, and the number of terms, being given, to find the sum of all the terms.

RULE II,

Multiply the sum of the extremes by the number of terms, and half the product will be the answer.

3. A lady purchased 19 yards of riband, for which she gave 1c. for the first yard, 3c. for the second, and 5c. for the third yard, increasing 2c. per yard; required the cost?

19 less 118

com. diff.

༤

い

X 19 no. of terms.

then 1+37=38

2X

36

2)722

last term, 37

half product, D3.61 Ans.

4. If 100 stones were laid two yards distant from each other, in a right line, and a basket placed two yards from the first stone; what distance would a person travel, to gather them singly into a basket? Ans. 11 m., 3 fur. 180 yds.

NOTE. In this question, there being 1760 yards in a mile, and the man returning with each stone to the basket, his travel will be doubled.

Given the extremes and the common difference, to find the number of terms.

RULE III.

Divide the difference of the extremes by the common difference, and the quotient, increased by 1, will be the number of terms required.

5. The extremes are 3 and 39, and the common difference 2; what is the number of terms?

39-3-36÷2 com. diff. 18 quot.+1=19. Ans. 6. A man going a journey travelled the first day 7 miles, the last day 51 miles, and each day increased his journey by 4 miles ; how many days did he travel, and how far? Ans. 12 days; 348 m. The extremes and common difference given, to find the sum of all the series.

RULE IV.

Multiply the sum of the extremes by their difference, increased by the common difference, and the product, divided by twice the common difference, will give the sum.

7. If the extremes are 3 and 39, and the common difference 2, what is the sum of the series? 39+3=42 sum of the extremes; 39-3=36, difference of extremes; 36+2=38, difference of extremes increased by the common divisor; 42×38= 15964, twice the common difference 399. Ans.

=

The extremes and sum of the series given, to find the number of

terms.

RÜLE V.

Twice the sum of the series, divided by the sum of the extremes, will give the number of terms.

8. Let the extremes be 3 and 39, and the sum of the series 399; what is the number of terms? Sum of the series-399 x2=798; then sum of the extremes 19. Ans.

39+3=42: 798÷42

The extremes and the sum of the series given, to find the common

difference.

RULE VI.

Divide the product of the sum and the difference of the extremes by the difference of twice the sum of the series and the sum of the extremes, and the quotient will be the common difference.

1

9. Let the extremes be 3 and 39, and the sum 399; what is he common difference? Sum.-Extremes=39+3=42; difference of extremes=39-3=36; 42×36=1512; then 399 X2-42-756)1512(2 Ans.

The first term, the common difference, and the number of terms given, to find the last term.

RULE VII.

The number of terms, less 1, multiplied by the common difference, and the first term added to the product, will give the last term.

10. A man bought 100 yards of cloth, giving 4c. for the first yard, 7c. for the second, 10c. for the third, and so on, with a common difference of 3c.; what was the cost of the last yard? No. of terms, 100-1=99×3, com. diff.=297+4 the first term 3.01D. Ans.

11. A man travelled 20 miles the first day, 24 the second, and so on, increasing 4 miles every day; how far did he travel the 12th day? Ans. 64 m.

NOTE. The great variety of cases in arithmetical progression will not permit more space to be occupied in this work; such questions as usually occur may be solved by the above rules.

12. A man had 8 sons, whose ages differed alike; the youngest was 10 years old, and the eldest 45; what was the common difference of their ages ? Ans. 5 years.

13. A man is to travel from New York to a certain place in 19 days, and to go but six miles the first day, increasing every day by an equal excess, so that the last day's journey may be 60 miles; what is the common difference, and distance of the journey? Ans. com. diff. 3 miles; distance 627 miles. 14. It is required to know how many times the hammer of a clock would strike in a week, or 168 hours, provided it increases 1 at each hour?

Ans. 14196.

15. What will D1 at 6 per cent. amount to in 20 years, at simple interest?

Ans. D2.20.

16. How many times does a regular clock strike in 12 hours?

Ans. 78.

17. A man bought 100 oxen, and gave for the first ox D1; for the second, D2; for the third, D3; and so on to the last; how much did they come to? Ans. 5050.

REVIEW.

What is Arithmetical Progression? Name the five things that should be particularly attended to in this rule? How do you

form an arithmetical series? What is the common difference? What is an ascending series? What is a descending series? What are the several numbers called? What are the first and last terms called? Ans. Extremes, and the intermediate terms are called the means. How do you find the common difference, when you know the two extremes and number of terms? What is rule 1st? 2d? &c.

18. If a piece of land, 60 rods in length, be 20 rods wide at one end, and at the other terminate in an angle or point, what number of square rods does it contain? Ans. 600.

GEOMETRICAL PROGRESSION.

GEOMETRICAL Progression is a series of numbers increasing or decreasing by a common ratio; thus, 2, 4, 8, 16, 32, increased by the multiplier or ratio 2, and 32, 16, 8, 4, 2, decrease continually by the divisor 2, &c. In this rule there are five denominations, any three of which being given, the other two may be found: 1st, the first term; 2d, the last term; 3d, the number of terms; 4th, the ratio; 5th, the sum of the series. The ratio is the multiplier or divisor by which the series is founded. To raise a power or series of numbers by the ratio, we place the ratio at the left hand for the first power; this (first power) multiplied by the ratio (its square) gives the second power, the second by the ratio gives the third, and so on, until the power is one number less than the first term. Let 2, 4, 8, 16, 32, &c., be the series whose ratio is 2. The second term is formed by multiplying the first term by the ratio 2; the third term by multiplying the second by the ratio, and so on. The series may therefore be written thus: 2, 2×2, 2×2×2, 2×2×2×2, 2× 2×2×2×2, &c., or thus: 2, 2 × 21, 2 × 22, ́2 × 2.3, 2 × 2a, &c. Any power after the first is evidently that power of the ratio whose index is one less than the number of the term multiplied by the first term; thus the third term is 2×22, the 4th term is 2×23, and the 8th term would be 2 × 27, &c. In an ascending series, therefore, multiply the first term by that power of the ratio whose index is one less than the number of the term sought, as mentioned above, and the product is the term sought. In a descending series, as 243, 81, 27, 9, 3, 1, whose ratio is 3, and which is also 1x35, 1x34, 1x33, 1x32, 1x31, 1,