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Questions. 1. What is Alligation ?

limited to a certain quantity, how do 2. What is Alligation Medial?

you find what quantity of each of the 3. What is the rule ?

others must be taken in proportion to 4. What is Alligation Alternate ? the given quantity?

5. When the mean rate and the rates 7. When the whole composition is of the ingredients are given without limited to a given quantity, how do you any limited quantity, how do you find find the proportional quantities of each the proportional quantity ?

ingredient? 6. when one of the ingredients is

ARITHMETICAL PROGRESSION.

Any rank, or series of numbers more than two, increas, ing by a common excess, or decreasing by a common difference, is called an Arithmetical Series or Progression.

The number which is continually added or subtracted, is called the common difference.

When the numbers are formed by a continual addition of the common difference it is called an ascending series ; but when they are formed by a continual subtraction of the common difference, they form a descending series. Thus,

$ 2, 4, 6, 8, 10, is an ascending series.

10, 8, 6, 4, 2, is a descending series. The numbers which form the series are called the terms of the series, or progression; the first and last terms of which are called the extremes.

A series in progression includes five parts, viz. :
1. The first term.
2. The last term.
3. The number of terms.
4. The common difference.

5. The sum of all the terms : by having any three of which given, the other two may be found.

CASE I.

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

RULE.

Multiply the number of terms, less 1, by the common difference, and to the product add the first term, and the sum will be the last term.

X3

EXAMPLES. 1. A man bought 17 yards of cloth, giving 5 cents for the first yard, 8 cents for the second, 11 cents for the third, and so on, increasing with a common difference of three cents on each yard ; what was the cost of the last yard ?

It will be seen that the Numb. of terms, less 1,=16

common difference will always be added one time less

than the number of terms.

48 First term,

+5

Thus, for the second yard the common difference is

added to the first yard one Last term,

-53

time, for the third yard it is added 2 times; and for the 17th yard, it must be added to the price of the first yard 16 times. 16X3=48, that is the 17th yard will cost 48 cents more than the 1st yard ; and 48+5=53, the price of the last.

2. If the first term be 5, and the common difference 3, and the number of terms 100, what is the last term ?

Ans. 302, 3. A man, in traveling a certain journey in 11 days, traveled thirteen miles the first day, and increased every day two miles; how many miles did he travel the last day?

Ans. 33 miles.

CASE II.

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

RULE.

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

EXAMPLES.

1. A man bought 17 yards of cloth in arithmetical progression. For the first yard he gave 5 cents, and for the last 53 cents ; what was the common difference, or the increase of the price on each succeeding yard ?

Extremes < 25

This operation is exactly the reverse of

those in Case I, and No. of terms, $ 16)48(3 com. diff.

taking the 1st exless 1,

treme from the last ev. idently shows the whole increase of the 16 additions ; and dividing by 16, the number of terms, less 1, shows the in: crease of one addition ; that is, the common difference.

2. If the extremes be 3 and 273, and the number of terms 46, what is the common difference ?

Ans. 6. 3. A man had 8 sons whose several ages differed alike ; the youngest was 7 years old and the oldest 35; what was the common difference of their ages ?

Ans. 4yrs. 4. A man performed a journey in 11 days. The first day he traveled thirteen miles and the last day 33, increasing every day by an equal excess; what was the daily increase ?

Ans. Daily increase 2 miles.

CASE III.

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

RULE.

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

EXAMPLES.

1. A man bought 12 yards of cloth in arithmetical pro: gression. For the first yard he gave 6 cents, and for the last yard 39 cents ; what did the whole amount to? Operation.

If we take half the price 6

of the first and last yards it 39

will give the average price

of the whole. Thus, 6+39 Sum of the extremes 45

=45, the sum of the exNumber of terms

X 12

tremes, and 45--2=22cts.

the average price, and 221 )540

12=270cts., the Ans.

But if we multiply the 27Qc.

whole sum of the extremes by the number of terms, (as in the operation,) and take half

the product, it will produce the same effect, and will generally be found preferable, as it will prevent the necessity of fractions. Hence the Rule.

2. If the first term of an arithmetical series be 3, the last term 150, and the number of terms 50, what is the sum of the series ?

Ans. 3825. 3. How many strokes does a regular clock strike in 12 hours ?

Ans. 78. 4. What is the sum of the first 25 numbers in their nat. yral order ? thus, 1, 2, 3, 4, 5, &c.

Ans. 325. 5. A merchant sold 100 yards of cloth. For the first yard he received 12 cents, for the second 15cts., for the ihird 18 cents, and so on, increasing 3 cents on each succeeding yard to the last; what did he receive for the last yard, and what was the amount of the whole ?

(Note. We must find the last term by the Rule, Case I. Thus, 3 x 99+12=309.)

Ans. Last yard $3,09; whole amount $160,50. 6. A man performed a journey in 11 days, traveling 13 miles the 1st day, and increasing every day by an equal excess until the last day's travel was 33 miles : what was the daily increase, and the whole distance traveled ?

(The daily increase or common difference is found by Case 2.)

Ans. Daily increase 2m., whole distance 253m. 7. If the first term in an arithmetical progressien be 4, the second 7, the third 10, &c. up to the 81st term, what is the sum of all the terms ?

Ans. 10044. 8. If 140 oranges be placed in a straight line 2 yards distant from each other, and a basket placed 2 yards dis: tant from the first orange, what distance must that boy travel who gathers them up singly, returning with them í by 1 to the basket ?

(Obs. It will be seen that the travel to the 1st orange and back again to the basket will be 2+2=4 yards, and to the last orange and back again to the basket will be 4 x 140 =560 yards. Hence the first term is 4, the last term 560, and the number of terms 140.)

Ans. 22 miles 3 furlongs 100 yards.

CASE IV.

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

RULE.

less 1,

}

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

EXAMPLES. 1. A man bought cloth in arithmetical progression, giving 5 cents for the first yard, and increasing the price of each succeeding yard by 3 cents to the last, which was 53 cents; how many yards did he buy?

Extremes

53 {

We found (Case 2, Ex. 1,

-5 that the difference of the exCommon difference 3)48 tremes, divided by the numNumber of terms,

16

ber of terms, less 1,) gave

+1 the common difference, and Ans.

yards 17

this question being the re

verse, it is evident that the difference of the extremes, divided by the common difference, will give the number of terms, less 1.

2. If the extremes be 3 and 273, and the common differ: ence 6, what is the number of terms ?

Ans. 46. 3. A man being asked how many children he had, replied, that his youngest was 7 years old and his oldest 35, and the common difference of their ages was four years ; how many had he?

Ans. 8 4. A man going a journey traveled the first day thirteen miles, and increased every day's travel two miles to the last, which was 33 miles; how many days did he travel, and how far did he travel ?

Ans. 11 days, and the distance traveled 253 miles:

Questions 1. What is Arithmetical Progression ? 5. What are the numbers called which

2. What is the number which is con form the series? tinually added or subtracted called ? 6. What are the first and last terms

3. When the numbers are formed by called ? a continual addition of the common dif 7. How many parts does a series in ference, what is it called ?

progression include, and what are they? 4. When formed by a continual sub 8. Having the first term, common diftraction, what is it called?

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