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RULE.

Divide the sum of all the terms by half the sum of the extremes.

Examples.

1. The first term of an arithmetical progression is 1, the last term is 1001, and the sum of all the terms is 251001. What is the number of terms?

In this example half the sum of the extremes is

1001+1 2

501; then dividing the sum of all the terms by this, we obtain 21001-501, for the number of terms.

2. In a triangular field of corn, the number of hills in the successive rows are in arithmetical progression: in the first row there is but one hill, in the last row there is 81 hills; and the whole number of hills in the field is 1681. How many rows are there? Ans. 41.

3. A man bought a certain number of yards of cloth for $152.50, giving 4 cents for the first yard, and increasing regularly on each succeeding yard, up to the last yard, for which he gave $3.01. How many yards of cloth did he purchase? Ans. 100.

4. How many terms are there in an arithmetical progression whose first term is 5, last term 75, and sum of all the terms 440?

CASE VII.

We also infer from Case II., that the sum of all the terms, divided by half the number of terms, will give the sum of the extremes. Therefore if from the quotient, of the sum of all the terms, divided by half the number of terms, we subtract the last term, we shall have left the first term.

Hence, when we have given the last term, the number of

terms, and the sum of all the terms, to find the first term, we have this

RULE.

From the quotient of the sum of all the terms, divided by half the number of terms, subtract the last term.

Examples.

1. If the last term of an arithmetical progression is 170, the number of terms 50; and the sum of all the terms 4450, what is the first term?

In this example the sum of all the terms divided by half the number of terms is 440-178, from which subtracting the last term, we obtain 178-170-8, for the first term.

2. A person wishes to discharge a debt of 1125 dollars in 18 annual payments, which shall be in arithmetical progression. How much must his first payment be, so as to bring his last payment 120 dollars? Ans. 5 dollars.

3. What is the first term of an arithmetical progression whose number of terms is 27, last term 50, and sum of all the terms 729 ?

Ans. 4.

4. The miles which a person travels in 19 successive days form an arithmetical progression, whose last term is 80, the sum of all the terms 950. How many miles did he travel the first day?

CASE VIII,

From what has been said under Case VII., we infer that the first term subtracted from the quotient, of the sum of all the terms divided by half the number of terms, will give the last term.

Hence, when we have given the sum of all the terms, the

first term, and the number of terms, to find the last term, we have this

RULE.

From the quotient of the sum of all the terms, divided by half the number of terms, subtract the first term.

Examples.

1. If the first term of an arithmetical progression is 7, the number of terms 1000, and the sum of all the terms 560000, what is the last term?

In this example the sum of all the terms, divided by half the number of terms, gives 560000=1120, from which subtract the first term, we get 1120-7-1113, for the last term.

2. If the first term of an arithmetical progression is 7, the number of terms 16, and the sum of all the terms 142, what is the last term? Ans. 10.

3. The first term of an arithmetical progression is 13, the number of terms 100, and the sum of all the terms 50300. What is the last term?

NOTE. The remaining cases are obtained by combining the conditions of Cases I. and II.

CASE IX.

Given the first term, the common difference, and the sum of all the terms, to find the last term.

RULE.

To twice the product of the common difference into the sum of all the terms, add the square of the first term, diminished by half the common difference; extract the square root of the sum ; from this root subtract half the common difference.

Examples.

1. The first term of an arithmetical progression is 4, the common difference is 7, and the sum of all the terms is 10233. What is the last term?

In this example, twice the product of the sum of all the terms into the common difference is 2 × 10233 ×7=143262; the square of the first term diminished by half the common difference is (4-7)(); this added to 143262 gives 573040, whose square root is 77; from this subtracting half the common difference we get 757—7—750—375, for the last

term.

2. The first term of an arithmetical progression is, the common difference is, and the sum of all the terms is 1180. What is the last term? Ans. 391.

3. A man has several sons, whose ages are in arithmetical progression; the age of the youngest is 5 years, the common difference of their ages is 6 years, and the sum of all their ages is 161. What is the age of the eldest?

Ans. 41 years.

4. The first term of an arithmetical progression is 17, the common difference is 4, and the sum of all the terms is 1142288. What is the last term?

CASE X.

Given the common difference, the number of terms, and the sum of all the terms, to find the last term:

RULE.

To the quotient of the sum of the terms, divided by the num ber of terms, add half the product of the common difference into the number of terms, less one.

Examples.

1. The common difference of the terms of an arithmetical progression is 6, the number of terms is 7, and the sum of all the terms is 161. What is the last term?

In this example, the sum of the terms divided by the number of terms is 11=23. 1 Again, the common difference multiplied into the number of terms, less one, is 6×6=36, the half of which is 18, which added to 23, gives 41, for the last term.

2. The common difference of the terms of an arithmetical progression is 7, the number of terms is 54, and the sum of all the terms is 10233. What is the last term? Ans. 375.

3. The common difference of the terms of an arithmetical progression is 6, the number of terms is 14, and the sum of all the terms is 4970.

What is the last term?

Ans. 394.

4. The common difference of the terms of an arithmetical progression is 17, the number of terms is 48, and the sum of What is the last term?

all the terms is 38496.

CASE XI.

Given the first term, the common difference, and the number of terms, to find the sum of all the terms.

RULE.

To twice the first term, add the product of the common difference into the number of terms, less one; multiply this sum by half the number of terms.

Examples.

1. The first term of an arithmetical progression is 37, the common difference is 11, and the number of terms 99. What is the sum of all the terms?

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